Name each polygon by its number of sides.
Then classify it as convex or concave and
regular or irregular.
1.
SOLUTION:
The polygon has 5 sides. A polygon with 5 sides is a
pentagon.
If you draw the line containing the side itwill
contain some points in the interior of the polygon. So,
the polygon is concave. Since it is concave, it is
irregular.
2.
SOLUTION:
The polygon has 9 sides. A polygon with 9 sides is a
nonagon.
None of the lines containing the sides will have points
in the interior of the polygon. So, the polygon is
convex.
All sides of the polygon are congruent and all angles
are congruent. So it is regular.
SIGNS Identify the shape of each traffic sign
and classify it as regular or irregular.
3.stop
SOLUTION:
Stop signs are constructed in the shape of a polygon
with 8 sides of equal length. The polygon has 8 sides.
A polygon with 8 sides is an octagon. All sides of the
polygon are congruent and all angles are also
congruent. So it is regular.
4.caution or warning
SOLUTION:
Caution and warning signs are constructed in the
shape of a polygon with 4 sides of equal length.
The polygon has 4 sides. A polygon with 4 sides is a
quadrilateral.
All sides of the polygon are congruent. So it is
regular.
5.slow moving vehicle
SOLUTION:
Slow moving vehicle signs are constructed in the
shape of a polygon with 6 sides of alternating length.
The polygon has 6 sides. A polygon with 6 sides is a
hexagon. All sides of the polygon are not congruent.
So it is irregular.
Find the perimeter or circumference and area of
each figure. Round to the nearest tenth.
6.
SOLUTION:
The perimeter of a square with side s is given by
.
Substitute 11 for s.
The perimeter of the figure is 44 ft.
The area of a square with side s is given by A = s2.
Substitute 11 for s.
The area of the square is 121 ft2.
7.
SOLUTION:
The circumference of a circle with radius r is given
by C = 2πr.
The diameter of the circle is 12.8 cm.
Substitute 6.4 for r.
The circumference of the circle is about 40.2 cm.
The area of a circle with radius r is given by A = πr2.
Substitute 6.4 for r.
The area of the circle is about 128.7 cm2.
8.
SOLUTION:
Add all the sides to find the perimeter of a triangle.
The perimeter of the triangle is 40 m.
The area of a triangle with base b and height h is
given by .
Here the base is 8 and height is 15.
The area of the triangle is 60 m2.
9.MULTIPLE CHOICE Vanesa is making a banner
for the game. She has 20 square feet of fabric. What
shape will use most or all of the fabric?
A asquarewithasidelengthof4feet
B arectanglewithalengthof4feetandawidthof
3.5 feet
C acirclewitharadiusofabout2.5feet
D arighttrianglewithlegsofabout5feet
SOLUTION:
The area of a square with side length 4 ft is
Area of square is 16 ft2.
The area of a rectangle with a length of 4 feet and a
width of 3.5 feet is
Area of rectangle is 14 ft2.
Theareaofacirclewithradius2.5ftis
The area of the circle is about 19.6 ft2.
Theareaofarighttrianglewithlegsofabout5ftis
The area of the triangle is 12.5 ft2.
So, the shape which uses the most of the fabric is the
circle.
The correct answer is C.
10.CCSS REASONING Find the perimeter and area
of withverticesA(–1, 2), B(3, 6), and C(3, –
2).
SOLUTION:
Graph .
To find the perimeter of , first find the lengths
ofeachside.
Use the Distance Formula to find the lengths of
.
has end points A(–1, 2) and B(3,6).
has endpoints B(3, 6) and C(3, –2).
has endpoints A(–1, 2) and C(3, –2).
The perimeter of is or
Find the area of .
To find the area of , find the lengths of the
height and base. The height of the triangle is the
horizontal distance from A to .
Counting the squares on the graph, the height is 4
units. The length of is 8 units.
The area of is 16 square units.
Name each polygon by its number of sides.
Then classify it as convex or concave and
regular or irregular.
11.
SOLUTION:
The polygon has 3 sides, so it is a triangle.
None of the lines containing the sides have points in
the interior of the polygon. So, the polygon is convex.
(All triangles are convex.)
All sides of the polygon are congruent and all angles
are also congruent. So it is regular.
12.
SOLUTION:
The polygon has 7 sides, so it is a heptagon.
Some of the lines containing the sides will have points
in the interior of the polygon. So, the polygon is
concave.
Since the polygon is concave, it is irregular.
13.
SOLUTION:
The polygon has 8 sides, so it is a octagon.
At least one of the lines containing the sides will have
points in the interior of the polygon. So, the polygon is
concave.
Since it is concave, it is irregular.
14.
SOLUTION:
The polygon has 4 sides, so it is a quadrilateral.
None of the lines containing the sides will have points
in the interior of the polygon. So, the polygon is
convex.
All sides of the polygon are not congruent. So it is
irregular.
15.
SOLUTION:
The polygon has 11 sides, so it is an hendecagon.
Al lease one of the lines containing the sides will
have points in the interior of the polygon. So, the
polygon is concave. Since it is concave, it is irregular.
16.
SOLUTION:
The polygon has 5 sides, so it is a pentagon.
None of the lines containing the sides will have points
in the interior of the polygon. So, the polygon is
convex.
All sides of the polygon are congruent and all angles
are also congruent. So it is regular.
Find the perimeter or circumference and area of
each figure. Round to the nearest tenth.
17.
SOLUTION:
Use the formula for perimeter of a rectangle.
The perimeter of the rectangle is 7.8 m.
Use the formula for area of a rectangle.
The area of the rectangle is about 3.1 m2.
18.
SOLUTION:
Use the formula for circumference of a circle.
The diameter of the circle is 8 in. So, the radius of
the circle is 4 in.
The circumference of the circle is about 25.1 in.
Use the formula for area of a circle.
The area of the circle is about 50.3 in2.
19.
SOLUTION:
Use the formula for perimeter of a square.
The perimeter of the square is 26 in.
Use the formula for area of a square.
The area of the square is about 42.3 in2.
20.
SOLUTION:
Add all the sides to find the perimeter of a triangle.
The perimeter of the triangle is 36 ft.
The area of a triangle with base b and height h is
given by .
Here the base is 5 ft and height is 8 ft.
The area of the triangle is 20 in2.
21.
SOLUTION:
To find the missing length, use the Pythagorean
Theorem.
Let c be the missing length.
Then,
Add all the sides to find the perimeter of a triangle.
The perimeter of the triangle is about 18.9 cm.
The area of a triangle with base b and height h is
given by .
Here the base is 4.5 ft and height is 6.5 ft.
The area of the triangle is about 14.6 cm2.
22.
SOLUTION:
Use the formula for circumference of a circle.
The circumference of the circle is about 36.4 cm.
Use the formula for area of a circle.
The area of the circle is about 105.7 cm2.
23.CRAFTS Joy has a square picture that is 4 inches
on each side. The picture is framed with a length of
ribbon. She wants to use the same piece of ribbon to
frame a circular picture. What is the maximum radius
of the circular frame?
SOLUTION:
Find the perimeter of the square picture.
Use the formula for the perimeter of a square with
side s.
The perimeter of the square picture is 16 inches.
So, the length of the ribbon is 16 inches.
If the picture is circular, then its circumference is 16
inches.
Use the circumference formula to solve for r.
The maximum radius of circular frame should be
about 2.55 in.
24.LANDSCAPING Mr. Jackson has a circular
garden with a diameter of 10 feet surrounded by
edging. Using the same length of edging, he is going
to create a square garden. What is the maximum side
length of the square?
SOLUTION:
The diameter of the garden is 10 feet. So the radius
is 5 feet.
To find the length of the edge, find the circumference
of the circular garden.
The length of the edging is about 31.4 ft.
Now, 31.4 is the perimeter of the square garden.
Equate it to 4s and solve for s.
The maximum side length of the square is about 7.85
ft.
CCSS REASONING Graph each figure with the
given vertices and identify the figure. Then find
the perimeter and area of the figure.
25.D(–2, –2), E(–2, 3), F(2, –1)
SOLUTION:
Graph the figure.
The polygon has 3 sides. So, it is a triangle.
To find the perimeter of , first find the lengths
of each side. Counting the squares on the grid, we
find that ED = 5.
Use the Distance Formula to find the lengths of
.
has end points E(–2, 3) and F(2, –1).
Substitute.
has end points D(–2, –2) and F(2, –1).
The perimeter of is
Find the area of .
To find the area of , find the lengths of the
height and base. The height of the triangle is the
horizontal distance from F to . Counting the
squares on the graph, the height is 4 units. The length
of is 5 units.
The area of is 10 square units.
26.J(–3, –3), K(3, 2), L(3, –3)
SOLUTION:
Graph the figure.
The polygon has 3 sides. So, it is a triangle.
To find the perimeter of , first find the lengths
of each side. Counting the squares on the grid, we
find that JL = 6 and KL = 5.
Use the Distance Formula to find the length of .
has end points J(–3, –3) and K(3, 2).
The perimeter of isHK + KL + JK.
Find the area of .
Here the base is 6 and the height is 5.
The area of is 15 square units.
27.P(–1, 1), Q(3, 4), R(6, 0), S(2, –3)
SOLUTION:
Graph the figure.
The polygon has 4 sides. So it is a quadrilateral.
To find the perimeter of the quadrilateral, find the
length of each side.
Use the Distance Formula to find the lengths of each
side.
has end points P(–1, 1) and Q(3, 4).
has end points Q(3, 4) and R(6, 0).
has end points R(6, 0) and S(2, –3).
has end points P(–1, 1) and S(2, –3).
Note that all the sides are congruent. Using a
protractor, all four angles of the quadrilateral are
right angles. So, it is a square.
Use the formula for the perimeter of a square with
sides of length s.
The perimeter of the square is 20 units.
Use the area formula for a square with sides of
length s.
The area of the square is 25 square units.
28.T(–2, 3), U(1, 6), V(5, 2), W(2, –1)
SOLUTION:
Graph the figure.
The polygon has 4 sides. So it is a quadrilateral.
To find the perimeter of the quadrilateral, find the
length of each side.
Use the Distance Formula to find the lengths of each
side.
has end points T(–2, 3) and U(1, 6).
has end points U(1, 6) and V(5, 2).
has end points V(5, 2) and W(2, –1).
has end points T(–2, 3) and W(2, –1).
Note that opposite sides are congruent. Using a
protractor, all the angles are right angles. So, the
quadrilateral is a rectangle.
Use the formula for the perimeter of a rectangle with
length and width w.
The perimeter of the rectangle is about 19.8 units.
Use the area formula for a rectangle with length
and width w.
The area of the rectangle is 24 square units.
29.CHANGING DIMENSIONS Use the rectangle
below.
a. Find the perimeter of the rectangle.
b. Find the area of the rectangle.
c. Suppose the length and width of the rectangle are
doubled. What effect would this have on the
perimeter? the area? Justify your answer.
d. Suppose the length and width of the rectangle are
halved. What effect does this have on the perimeter?
the area? Justify your answer.
SOLUTION:
a. Use the formula for the perimeter of a rectangle
with length and w.
The perimeter of the rectangle is 14 ft.
b. Use the formula for the area of a rectangle with
length and width w.
The area of the rectangle is 12 ft2.
c. If the length and width of the rectangle are
doubled, then the dimensions of the rectangle are 6 ft
and 8 ft. The perimeter of a rectangle with
dimensions 6 ft and 8 ft is 2(6+8) or 28 ft, which is
twice the perimeter of the original figure since
So,ifthelengthandwidthoftherectangle
are doubled, then the perimeter also doubled. The
area of the rectangle with dimensions 6 ft and 8 ft is
48 ft2, which is 4 times the original figure since
. So, the area quadruples.
d. If the length and width of the rectangle are halved,
then the dimensions of the rectangle are 1.5 ft and 2
ft. The perimeter of a rectangle with dimensions 1.5
ft and 2 ft is 2(1.5 + 2) or 7 ft, which is half the
perimeter of the original figure since .
So, if the length and width of the rectangle are
halved, then the perimeter also halved.
The area of the rectangle with dimensions 1.5 ft and
2 ft is 3 ft2, which is timestheoriginalfiguresince
.
So, the area is divided by 4.
30.CHANGING DIMENSIONS Use the triangle
below.
a. Find the perimeter of the triangle.
b. Find the area of the triangle.
c. Suppose the side lengths and height of the triangle
were doubled. What effect would this have on the
perimeter? the area? Justify your answer.
d. Suppose the side lengths and height of the triangle
were divided by three. What effect would this have
on the perimeter? the area? Justify your answer.
SOLUTION:
a.
The perimeter of the triangle is 33 m.
b. The area of the rectangle is:
The area of the rectangle is 27 m2.
c. If the sides and height are doubled, then the sides
of the triangle are 24, 12, and 30 and the height is 18.
Its perimeter is 24 + 12 + 30 or 66 m. This is twice
the perimeter of the original figure since 2(33) = 66.
So, the perimeter also doubles.
Its area is or 108 m2. This is four times the
area of the original figure since 4(27) = 108. The
area of the triangle quadruples.
d. If the side lengths and height were divided by 3,
then the side lengths are 4, 2, and 5 and the height is
3.
The perimeter is 4 + 2 + 5 or 11 m. This is ofthe
perimeter of the original figure since . So,
the perimeter is divided by 3. The area is or 3
m2. This is oftheareaoftheoriginalfiguresince
.
So, the area is divided by 9.
31.ALGEBRA A rectangle of area 360 square yards is
10 times as long as it is wide. Find its length and
width.
SOLUTION:
Let x be the width. Then the length is 10x.
Use the area formula for a rectangle.
Since width can never be negative, x = 6.
The length of the rectangle is 10x = 10(6) = 60 yards
and the width of the rectangle is 6 yards.
32.ALGEBRA A rectangle of area 350 square feet is
14 times as wide as it is long. Find its length and
width.
SOLUTION:
Let x be the length. Then the width is 14x.
Use the area formula for a rectangle. Equate it to
350 and solve for x
Sincelengthcanneverbenegative,x = 5.
The length of the rectangle is 5 ft and the width is
14x=14(5)=70ft.
33.DISC GOLF The diameter of the most popular
brand of flying disc used in disc golf measures
between 8 and 10 inches. Find the range of possible
circumferences and areas for these flying discs to
the nearest tenth.
SOLUTION:
The circumference is minimized when the diameter is
8 inches.
The minimum circumference is about 25.1 in.
The circumference is maximized when the diameter
is 10 in.
The maximum circumference is about 31.4 in.
The area is minimum when the radius is 4 inches.
The minimum area is about 50.3 in2.
The circumference maximum when the radius is 5 in.
The maximum area is about 78.5 in2.
ALGEBRA Find the perimeter or
circumference for each figure described.
34.The area of a square is 36 square units.
SOLUTION:
Find the length of the side.
Use the formula for the area of a square with side s.
Since the length can never be negative, s = 6.
Use the formula for perimeter of the square with side
s.
The perimeter of the square is 24 units.
35.The length of a rectangle is half the width. The area
is 25 square meters.
SOLUTION:
Let w be the width. So, the length of the rectangle is
.
Use the area formula for a the rectangle.
Therefore, the length is or 3.5.
Usetheformulaforperimeterofarectangle.
The perimeter of the rectangle is about 21.2 m.
36.The area of a circle is 25πsquare units.
SOLUTION:
Use the area formula for a circle with radius r.
The radius of the circle is 5 units.
Find the circumference.
Use the formula for the circumference of a circle
with radius r.
The circumference of the circle is 10πor about 31.4
units.
37.The area of a circle is 32πsquare units.
SOLUTION:
Use the formula for the area of a circle with radius r.
The radius of the circle is about units.
Find the circumference.
Use the formula for the circumference of a circle
with radius r.
The circumference of the circle is orabout
35.5 units.
38.A rectangle’s length is 3 times its width. The area is
27 square inches.
SOLUTION:
Let w be the width. So, the length of the rectangle is
3w.
Use the formula for the area of a rectangle.
Therefore, the length is 3(3) or 9 in.
Substitute in the formula for perimeter.
The perimeter of the rectangle is 24 in.
39.A rectangle’s length is twice its width. The area is 48
square inches.
SOLUTION:
Let w be the width. So, the length of the rectangle is
2w.
Use the formula for the area of the rectangle.
Therefore, the length is in.
Substitute in the formula for perimeter.
The perimeter of the rectangle is
CCSSPRECISIONFindtheperimeterand
area of each figure in inches. Round to the
nearest hundredth, if necessary.
40.
SOLUTION:
Before finding the perimeter and area, you must first
find the lengths of the two missing sides of the right
triangle. In the diagram, it is indicated that the base
is congruent to the height, so b=2.5cm.
Use the Pythagorean Theorem to find the length of
the hypotenuse.
The perimeter of the triangle is the sum of the sides.
Use dimensional analysis to change centimeters to
inches.
The area of the triangle is half the product of the
base and the height.
Use dimensional analysis to change cm2 to in2.
So, the perimeter is about 3.42 in. and the area is
0.5 in2.
41.
SOLUTION:
Use the formulas to find the perimeter and area of
the square.
Use dimensional analysis to change from yards to
inches.
Use dimensional analysis to change yd2 to in2.
So, the perimeter is 108 in. and the area is 729 in2.
42.
SOLUTION:
Use the formulas to find the perimeter and area of
the rectangle.
Use dimensional analysis to change feet to inches.
Use dimensional analysis to change ft2 to in2.
So, the perimeter is 223.2 in. and the area is
2767.68 in2.
43.Multiple Representations Collect and measure
thediameterandcircumferenceoftenroundobjects
using a millimeter measuring tape.
a. Tabular Record the measures in a table as
shown.
b. Algebraic Compute the value to the nearest
hundredth for each object and record the result.
c. Graphical Make a scatter plot of the data with
d-values on the horizontal axis and C-values on the
vertical axis.
d. Verbal Find an equation for a line of best fit for
the data. What does this equation represent? What
does the slope of the line represent?
SOLUTION:
a-b. Sample answer: Make a copy of the table given
for the problem. Find ten circular objects, then
measure the diameter and circumference of each
object to the nearest tenth of a centimeter, and
record the results in the table. Divide the
circumference by the diameter for each object and
record the result in the table. For example, an object
has a diameter of 3 cm and a circumference of 9.4
cm. .
c. Choose a scale for your horizontal and vertical
axis that will contain all your diameters and
circumferences. Plot the ten points determined by
each pair of diameter and circumference measures.
d.Sample answer: Enter your data into a graphing
calculator. In the STAT menu, enter the diameters
into L1 and the circumferences into L2. Then in the
STAT menu choose CALC and the LinReg(ax +
b) function to find the equation for the regression
line.
Here, a≈3.14andb≈0,soanequationforalineof
best fit would be C = 3.14d; the equation represents
a formula for approximating the circumference of a
circle. The slope represents an approximation for pi.
44.WHICH ONE DOESN’T BELONG? Identify the
term that does not belong with the other three.
Explain your reasoning.
SOLUTION:
Circle; The other shapes are polygons.
45.CHALLENGETheverticesofarectanglewith
side lengths of 10 and 24 are on a circle of radius
13 units. Find the area between the figures.
SOLUTION:
Startbydrawingthefigure.
The shaded region of the drawing represents the
area between the figures. Next, find the area of
each figure.
Use the formula to find the area of the rectangle
(Area of a rectangle = bh).
units2
Use the formula to find the area of the circle (Area
of a circle ).
units2
Then subtract the area of the rectangle from the
area of the circle in order to find the area of the
shaded region.
Therefore, the area between the figures is about
290.93 units2.
46.REASONING Name a polygon that is always
regular and a polygon that is sometimes regular.
Explain your reasoning.
SOLUTION:
Square; by definition, all sides of a square are
congruentandallanglesmeasure90°,sotherefore
are congruent. Triangle; triangles can have all sides
and angles congruent, just two sides and angle pairs
congruent, or no sides or angles congruent.
47.OPEN ENDED Draw a pentagon. Is your pentagon
convex or concave? Is your pentagon regular or
irregular? Justify your answers.
SOLUTION:
Sample answer: The pentagon is convex, since no
points of the lines drawn on the edges are in the
interior. The pentagon is regular since all of the
angles and sides were constructed with the same
measurement, making them congruent to each other.
48.CHALLENGE A rectangular room measures 20
feet by 12.5 feet. How many 5-inch square tiles will
it take to cover the floor of this room? Explain.
SOLUTION:
Convert the dimensions from feet to inches.
Thelengthoftheroomis20×12or240inchesand
thewidthoftheroomis12.5×12or150inches.It
needs 240 5 = 48 columns of tiles and 150 5 = 30
rows of tiles to cover this space. So the number of
tilesneededis48×30or1440tiles.
49.WRITING IN MATH Describe two possible ways
that a polygon can be equiangular but not a regular
polygon.
SOLUTION:
Sample answer: If a convex polygon is equiangular
but not also equilateral, then it is not a regular
polygon. Likewise, if a polygon is equiangular and
equilateral, but not convex, then it is not a regular
polygon.
50.Find the perimeter of the figure.
A 17 cm
B 25 cm
C 28 cm
D 31 cm
SOLUTION:
The lengths of two sides are unknown.
The length of the base is 4 + 4 or 8 cm.
To find the length of the unknown vertical side,
subtract 3 from 6.
6 – 3 = 3
Add all the sides to find the perimeter.
The perimeter of the figure is 28 cm
The correct choice is C.
51.PROBABILITY In three successive rolls of a fair
number cube, Matt rolls a 6. What is the probability
of Matt rolling a 6 if the number cube is rolled a
fourth time?
F
G
H
J 1
SOLUTION:
Probability is defined as
.
The number of favorable outcomes is 1, and the total
number of outcomes is 6.
So, the probability of rolling a 6 = .
The correct choice is F.
52.SHORT RESPONSE Miguel is planning a party for
80 guests. According to the pattern in the table, how
many gallons of ice cream should Miguel buy?
SOLUTION:
From the pattern we see that for every 8 guests
Miguel needs 2 gallons of ice cream.
So,for8×10or80guestsMiguelneeds2×10or20
gallons of ice cream.
53.SAT/ACT A frame 2 inches wide surrounds a
painting that is 18 inches wide and 14 inches tall.
What is the area of the frame?
A 68 in2
B 84 in2
C 144 in2
D 252 in2
E 396 in2
SOLUTION:
With 2 inch wide frame, the dimensions of painting
with the frame becomes 22 inches by 18 inches.
Find the area of the painting with frame.
A =22·18
=396in2
Find the area of the painting with out the frame.
A =18·14
=252in2
To find the area of the frame, subtract the area of
the paint from the area of the paint with the frame.
396 – 252 = 144
The area of the frame is 144 in2.
The correct choice is C.
Determine whether each statement can be
assumed from the figure. Explain.
54. KJN is a right angle.
SOLUTION:
Yes; the symbol denotes that KJN is a right angle.
55. PLN NLM
SOLUTION:
It might appear that the angles are congruent but
there are no marks on the angles in the diagram to
indicate this. There is also nothing in the diagram that
would provide information to know that bisects
∠PLM. Because we do not know anything about the
measures of these two angles, we can not assume
that ∠PLN ∠NLM .
56. PNL and MNL are complementary.
SOLUTION:
No; we do not know whether MNP is a right
angle.
57. KLN and MLN are supplementary.
SOLUTION:
Yes; they form a linear pair.
58.TABLE TENNIS The diagram shows the angle of
play for a table tennis player. If a right-handed player
has a strong forehand, he should stand to the left of
the center line of his opponent’s angle of play.
a. What geometric term describes the center line?
b. If the angle of play shown in the diagram
measures43°,whatis mBAD?
SOLUTION:
a. The center line divides the angle into two
congruent angles. So, it is an angle bisector.
b. Since the center line is an angle bisector,
Name an appropriate method to solve each
system of equations. Then solve the system.
59.
SOLUTION:
The appropriate method to solve this system is the
elimination method, since neither of the equations
have variables with coefficients of 1 and equations
can not be simplified.
Multiply the first equation by 2 and the second
equation by 5
Substitute the value of y in one of the given
equations.
The solution of the system is: x = –3, y = –1
60.
SOLUTION:
The appropriate method to solve this system is
graphing since both equations are in slope-intercept
form and can be easily graphed.
Graph the lines on a coordinate grid.
The graphs appear to intersect at (3, –8).
So, the solution of the system of equations is x = 3, y
= –8.
61.
SOLUTION:
The appropriate method to solve this system is
substitution, since equation 1 has a variable with a
coefficient of 1.
Solve the first equation for x.
Substitute this in the second equation.
Substitute the value of y in one of the given equation.
The solution to the system is x = –4, y = –2.5
Evaluate each expression if P = 10, B = 12, h =
6, r = 3, and =5.Roundtothenearesttenth,
if necessary.
62.
SOLUTION:
Substitute.
63.
SOLUTION:
Substitute.
64.
SOLUTION:
Substitute.
65.
SOLUTION:
Substitute.
Name each polygon by its number of sides.
Then classify it as convex or concave and
regular or irregular.
1.
SOLUTION:
The polygon has 5 sides. A polygon with 5 sides is a
pentagon.
If you draw the line containing the side itwill
contain some points in the interior of the polygon. So,
the polygon is concave. Since it is concave, it is
irregular.
2.
SOLUTION:
The polygon has 9 sides. A polygon with 9 sides is a
nonagon.
None of the lines containing the sides will have points
in the interior of the polygon. So, the polygon is
convex.
All sides of the polygon are congruent and all angles
are congruent. So it is regular.
SIGNS Identify the shape of each traffic sign
and classify it as regular or irregular.
3.stop
SOLUTION:
Stop signs are constructed in the shape of a polygon
with 8 sides of equal length. The polygon has 8 sides.
A polygon with 8 sides is an octagon. All sides of the
polygon are congruent and all angles are also
congruent. So it is regular.
4.caution or warning
SOLUTION:
Caution and warning signs are constructed in the
shape of a polygon with 4 sides of equal length.
The polygon has 4 sides. A polygon with 4 sides is a
quadrilateral.
All sides of the polygon are congruent. So it is
regular.
5.slow moving vehicle
SOLUTION:
Slow moving vehicle signs are constructed in the
shape of a polygon with 6 sides of alternating length.
The polygon has 6 sides. A polygon with 6 sides is a
hexagon. All sides of the polygon are not congruent.
So it is irregular.
Find the perimeter or circumference and area of
each figure. Round to the nearest tenth.
6.
SOLUTION:
The perimeter of a square with side s is given by
.
Substitute 11 for s.
The perimeter of the figure is 44 ft.
The area of a square with side s is given by A = s2.
Substitute 11 for s.
The area of the square is 121 ft2.
7.
SOLUTION:
The circumference of a circle with radius r is given
by C = 2πr.
The diameter of the circle is 12.8 cm.
Substitute 6.4 for r.
The circumference of the circle is about 40.2 cm.
The area of a circle with radius r is given by A = πr2.
Substitute 6.4 for r.
The area of the circle is about 128.7 cm2.
8.
SOLUTION:
Add all the sides to find the perimeter of a triangle.
The perimeter of the triangle is 40 m.
The area of a triangle with base b and height h is
given by .
Here the base is 8 and height is 15.
The area of the triangle is 60 m2.
9.MULTIPLE CHOICE Vanesa is making a banner
for the game. She has 20 square feet of fabric. What
shape will use most or all of the fabric?
A asquarewithasidelengthof4feet
B arectanglewithalengthof4feetandawidthof
3.5 feet
C acirclewitharadiusofabout2.5feet
D arighttrianglewithlegsofabout5feet
SOLUTION:
The area of a square with side length 4 ft is
Area of square is 16 ft2.
The area of a rectangle with a length of 4 feet and a
width of 3.5 feet is
Area of rectangle is 14 ft2.
Theareaofacirclewithradius2.5ftis
The area of the circle is about 19.6 ft2.
Theareaofarighttrianglewithlegsofabout5ftis
The area of the triangle is 12.5 ft2.
So, the shape which uses the most of the fabric is the
circle.
The correct answer is C.
10.CCSS REASONING Find the perimeter and area
of withverticesA(–1, 2), B(3, 6), and C(3, –
2).
SOLUTION:
Graph .
To find the perimeter of , first find the lengths
ofeachside.
Use the Distance Formula to find the lengths of
.
has end points A(–1, 2) and B(3,6).
has endpoints B(3, 6) and C(3, –2).
has endpoints A(–1, 2) and C(3, –2).
The perimeter of is or
Find the area of .
To find the area of , find the lengths of the
height and base. The height of the triangle is the
horizontal distance from A to .
Counting the squares on the graph, the height is 4
units. The length of is 8 units.
The area of is 16 square units.
Name each polygon by its number of sides.
Then classify it as convex or concave and
regular or irregular.
11.
SOLUTION:
The polygon has 3 sides, so it is a triangle.
None of the lines containing the sides have points in
the interior of the polygon. So, the polygon is convex.
(All triangles are convex.)
All sides of the polygon are congruent and all angles
are also congruent. So it is regular.
12.
SOLUTION:
The polygon has 7 sides, so it is a heptagon.
Some of the lines containing the sides will have points
in the interior of the polygon. So, the polygon is
concave.
Since the polygon is concave, it is irregular.
13.
SOLUTION:
The polygon has 8 sides, so it is a octagon.
At least one of the lines containing the sides will have
points in the interior of the polygon. So, the polygon is
concave.
Since it is concave, it is irregular.
14.
SOLUTION:
The polygon has 4 sides, so it is a quadrilateral.
None of the lines containing the sides will have points
in the interior of the polygon. So, the polygon is
convex.
All sides of the polygon are not congruent. So it is
irregular.
15.
SOLUTION:
The polygon has 11 sides, so it is an hendecagon.
Al lease one of the lines containing the sides will
have points in the interior of the polygon. So, the
polygon is concave. Since it is concave, it is irregular.
16.
SOLUTION:
The polygon has 5 sides, so it is a pentagon.
None of the lines containing the sides will have points
in the interior of the polygon. So, the polygon is
convex.
All sides of the polygon are congruent and all angles
are also congruent. So it is regular.
Find the perimeter or circumference and area of
each figure. Round to the nearest tenth.
17.
SOLUTION:
Use the formula for perimeter of a rectangle.
The perimeter of the rectangle is 7.8 m.
Use the formula for area of a rectangle.
The area of the rectangle is about 3.1 m2.
18.
SOLUTION:
Use the formula for circumference of a circle.
The diameter of the circle is 8 in. So, the radius of
the circle is 4 in.
The circumference of the circle is about 25.1 in.
Use the formula for area of a circle.
The area of the circle is about 50.3 in2.
19.
SOLUTION:
Use the formula for perimeter of a square.
The perimeter of the square is 26 in.
Use the formula for area of a square.
The area of the square is about 42.3 in2.
20.
SOLUTION:
Add all the sides to find the perimeter of a triangle.
The perimeter of the triangle is 36 ft.
The area of a triangle with base b and height h is
given by .
Here the base is 5 ft and height is 8 ft.
The area of the triangle is 20 in2.
21.
SOLUTION:
To find the missing length, use the Pythagorean
Theorem.
Let c be the missing length.
Then,
Add all the sides to find the perimeter of a triangle.
The perimeter of the triangle is about 18.9 cm.
The area of a triangle with base b and height h is
given by .
Here the base is 4.5 ft and height is 6.5 ft.
The area of the triangle is about 14.6 cm2.
22.
SOLUTION:
Use the formula for circumference of a circle.
The circumference of the circle is about 36.4 cm.
Use the formula for area of a circle.
The area of the circle is about 105.7 cm2.
23.CRAFTS Joy has a square picture that is 4 inches
on each side. The picture is framed with a length of
ribbon. She wants to use the same piece of ribbon to
frame a circular picture. What is the maximum radius
of the circular frame?
SOLUTION:
Find the perimeter of the square picture.
Use the formula for the perimeter of a square with
side s.
The perimeter of the square picture is 16 inches.
So, the length of the ribbon is 16 inches.
If the picture is circular, then its circumference is 16
inches.
Use the circumference formula to solve for r.
The maximum radius of circular frame should be
about 2.55 in.
24.LANDSCAPING Mr. Jackson has a circular
garden with a diameter of 10 feet surrounded by
edging. Using the same length of edging, he is going
to create a square garden. What is the maximum side
length of the square?
SOLUTION:
The diameter of the garden is 10 feet. So the radius
is 5 feet.
To find the length of the edge, find the circumference
of the circular garden.
The length of the edging is about 31.4 ft.
Now, 31.4 is the perimeter of the square garden.
Equate it to 4s and solve for s.
The maximum side length of the square is about 7.85
ft.
CCSS REASONING Graph each figure with the
given vertices and identify the figure. Then find
the perimeter and area of the figure.
25.D(–2, –2), E(–2, 3), F(2, –1)
SOLUTION:
Graph the figure.
The polygon has 3 sides. So, it is a triangle.
To find the perimeter of , first find the lengths
of each side. Counting the squares on the grid, we
find that ED = 5.
Use the Distance Formula to find the lengths of
.
has end points E(–2, 3) and F(2, –1).
Substitute.
has end points D(–2, –2) and F(2, –1).
The perimeter of is
Find the area of .
To find the area of , find the lengths of the
height and base. The height of the triangle is the
horizontal distance from F to . Counting the
squares on the graph, the height is 4 units. The length
of is 5 units.
The area of is 10 square units.
26.J(–3, –3), K(3, 2), L(3, –3)
SOLUTION:
Graph the figure.
The polygon has 3 sides. So, it is a triangle.
To find the perimeter of , first find the lengths
of each side. Counting the squares on the grid, we
find that JL = 6 and KL = 5.
Use the Distance Formula to find the length of .
has end points J(–3, –3) and K(3, 2).
The perimeter of isHK + KL + JK.
Find the area of .
Here the base is 6 and the height is 5.
The area of is 15 square units.
27.P(–1, 1), Q(3, 4), R(6, 0), S(2, –3)
SOLUTION:
Graph the figure.
The polygon has 4 sides. So it is a quadrilateral.
To find the perimeter of the quadrilateral, find the
length of each side.
Use the Distance Formula to find the lengths of each
side.
has end points P(–1, 1) and Q(3, 4).
has end points Q(3, 4) and R(6, 0).
has end points R(6, 0) and S(2, –3).
has end points P(–1, 1) and S(2, –3).
Note that all the sides are congruent. Using a
protractor, all four angles of the quadrilateral are
right angles. So, it is a square.
Use the formula for the perimeter of a square with
sides of length s.
The perimeter of the square is 20 units.
Use the area formula for a square with sides of
length s.
The area of the square is 25 square units.
28.T(–2, 3), U(1, 6), V(5, 2), W(2, –1)
SOLUTION:
Graph the figure.
The polygon has 4 sides. So it is a quadrilateral.
To find the perimeter of the quadrilateral, find the
length of each side.
Use the Distance Formula to find the lengths of each
side.
has end points T(–2, 3) and U(1, 6).
has end points U(1, 6) and V(5, 2).
has end points V(5, 2) and W(2, –1).
has end points T(–2, 3) and W(2, –1).
Note that opposite sides are congruent. Using a
protractor, all the angles are right angles. So, the
quadrilateral is a rectangle.
Use the formula for the perimeter of a rectangle with
length and width w.
The perimeter of the rectangle is about 19.8 units.
Use the area formula for a rectangle with length
and width w.
The area of the rectangle is 24 square units.
29.CHANGING DIMENSIONS Use the rectangle
below.
a. Find the perimeter of the rectangle.
b. Find the area of the rectangle.
c. Suppose the length and width of the rectangle are
doubled. What effect would this have on the
perimeter? the area? Justify your answer.
d. Suppose the length and width of the rectangle are
halved. What effect does this have on the perimeter?
the area? Justify your answer.
SOLUTION:
a. Use the formula for the perimeter of a rectangle
with length and w.
The perimeter of the rectangle is 14 ft.
b. Use the formula for the area of a rectangle with
length and width w.
The area of the rectangle is 12 ft2.
c. If the length and width of the rectangle are
doubled, then the dimensions of the rectangle are 6 ft
and 8 ft. The perimeter of a rectangle with
dimensions 6 ft and 8 ft is 2(6+8) or 28 ft, which is
twice the perimeter of the original figure since
So,ifthelengthandwidthoftherectangle
are doubled, then the perimeter also doubled. The
area of the rectangle with dimensions 6 ft and 8 ft is
48 ft2, which is 4 times the original figure since
. So, the area quadruples.
d. If the length and width of the rectangle are halved,
then the dimensions of the rectangle are 1.5 ft and 2
ft. The perimeter of a rectangle with dimensions 1.5
ft and 2 ft is 2(1.5 + 2) or 7 ft, which is half the
perimeter of the original figure since .
So, if the length and width of the rectangle are
halved, then the perimeter also halved.
The area of the rectangle with dimensions 1.5 ft and
2 ft is 3 ft2, which is timestheoriginalfiguresince
.
So, the area is divided by 4.
30.CHANGING DIMENSIONS Use the triangle
below.
a. Find the perimeter of the triangle.
b. Find the area of the triangle.
c. Suppose the side lengths and height of the triangle
were doubled. What effect would this have on the
perimeter? the area? Justify your answer.
d. Suppose the side lengths and height of the triangle
were divided by three. What effect would this have
on the perimeter? the area? Justify your answer.
SOLUTION:
a.
The perimeter of the triangle is 33 m.
b. The area of the rectangle is:
The area of the rectangle is 27 m2.
c. If the sides and height are doubled, then the sides
of the triangle are 24, 12, and 30 and the height is 18.
Its perimeter is 24 + 12 + 30 or 66 m. This is twice
the perimeter of the original figure since 2(33) = 66.
So, the perimeter also doubles.
Its area is or 108 m2. This is four times the
area of the original figure since 4(27) = 108. The
area of the triangle quadruples.
d. If the side lengths and height were divided by 3,
then the side lengths are 4, 2, and 5 and the height is
3.
The perimeter is 4 + 2 + 5 or 11 m. This is ofthe
perimeter of the original figure since . So,
the perimeter is divided by 3. The area is or 3
m2. This is oftheareaoftheoriginalfiguresince
.
So, the area is divided by 9.
31.ALGEBRA A rectangle of area 360 square yards is
10 times as long as it is wide. Find its length and
width.
SOLUTION:
Let x be the width. Then the length is 10x.
Use the area formula for a rectangle.
Since width can never be negative, x = 6.
The length of the rectangle is 10x = 10(6) = 60 yards
and the width of the rectangle is 6 yards.
32.ALGEBRA A rectangle of area 350 square feet is
14 times as wide as it is long. Find its length and
width.
SOLUTION:
Let x be the length. Then the width is 14x.
Use the area formula for a rectangle. Equate it to
350 and solve for x
Sincelengthcanneverbenegative,x = 5.
The length of the rectangle is 5 ft and the width is
14x=14(5)=70ft.
33.DISC GOLF The diameter of the most popular
brand of flying disc used in disc golf measures
between 8 and 10 inches. Find the range of possible
circumferences and areas for these flying discs to
the nearest tenth.
SOLUTION:
The circumference is minimized when the diameter is
8 inches.
The minimum circumference is about 25.1 in.
The circumference is maximized when the diameter
is 10 in.
The maximum circumference is about 31.4 in.
The area is minimum when the radius is 4 inches.
The minimum area is about 50.3 in2.
The circumference maximum when the radius is 5 in.
The maximum area is about 78.5 in2.
ALGEBRA Find the perimeter or
circumference for each figure described.
34.The area of a square is 36 square units.
SOLUTION:
Find the length of the side.
Use the formula for the area of a square with side s.
Since the length can never be negative, s = 6.
Use the formula for perimeter of the square with side
s.
The perimeter of the square is 24 units.
35.The length of a rectangle is half the width. The area
is 25 square meters.
SOLUTION:
Let w be the width. So, the length of the rectangle is
.
Use the area formula for a the rectangle.
Therefore, the length is or 3.5.
Usetheformulaforperimeterofarectangle.
The perimeter of the rectangle is about 21.2 m.
36.The area of a circle is 25πsquare units.
SOLUTION:
Use the area formula for a circle with radius r.
The radius of the circle is 5 units.
Find the circumference.
Use the formula for the circumference of a circle
with radius r.
The circumference of the circle is 10πor about 31.4
units.
37.The area of a circle is 32πsquare units.
SOLUTION:
Use the formula for the area of a circle with radius r.
The radius of the circle is about units.
Find the circumference.
Use the formula for the circumference of a circle
with radius r.
The circumference of the circle is orabout
35.5 units.
38.A rectangle’s length is 3 times its width. The area is
27 square inches.
SOLUTION:
Let w be the width. So, the length of the rectangle is
3w.
Use the formula for the area of a rectangle.
Therefore, the length is 3(3) or 9 in.
Substitute in the formula for perimeter.
The perimeter of the rectangle is 24 in.
39.A rectangle’s length is twice its width. The area is 48
square inches.
SOLUTION:
Let w be the width. So, the length of the rectangle is
2w.
Use the formula for the area of the rectangle.
Therefore, the length is in.
Substitute in the formula for perimeter.
The perimeter of the rectangle is
CCSSPRECISIONFindtheperimeterand
area of each figure in inches. Round to the
nearest hundredth, if necessary.
40.
SOLUTION:
Before finding the perimeter and area, you must first
find the lengths of the two missing sides of the right
triangle. In the diagram, it is indicated that the base
is congruent to the height, so b=2.5cm.
Use the Pythagorean Theorem to find the length of
the hypotenuse.
The perimeter of the triangle is the sum of the sides.
Use dimensional analysis to change centimeters to
inches.
The area of the triangle is half the product of the
base and the height.
Use dimensional analysis to change cm2 to in2.
So, the perimeter is about 3.42 in. and the area is
0.5 in2.
41.
SOLUTION:
Use the formulas to find the perimeter and area of
the square.
Use dimensional analysis to change from yards to
inches.
Use dimensional analysis to change yd2 to in2.
So, the perimeter is 108 in. and the area is 729 in2.
42.
SOLUTION:
Use the formulas to find the perimeter and area of
the rectangle.
Use dimensional analysis to change feet to inches.
Use dimensional analysis to change ft2 to in2.
So, the perimeter is 223.2 in. and the area is
2767.68 in2.
43.Multiple Representations Collect and measure
thediameterandcircumferenceoftenroundobjects
using a millimeter measuring tape.
a. Tabular Record the measures in a table as
shown.
b. Algebraic Compute the value to the nearest
hundredth for each object and record the result.
c. Graphical Make a scatter plot of the data with
d-values on the horizontal axis and C-values on the
vertical axis.
d. Verbal Find an equation for a line of best fit for
the data. What does this equation represent? What
does the slope of the line represent?
SOLUTION:
a-b. Sample answer: Make a copy of the table given
for the problem. Find ten circular objects, then
measure the diameter and circumference of each
object to the nearest tenth of a centimeter, and
record the results in the table. Divide the
circumference by the diameter for each object and
record the result in the table. For example, an object
has a diameter of 3 cm and a circumference of 9.4
cm. .
c. Choose a scale for your horizontal and vertical
axis that will contain all your diameters and
circumferences. Plot the ten points determined by
each pair of diameter and circumference measures.
d.Sample answer: Enter your data into a graphing
calculator. In the STAT menu, enter the diameters
into L1 and the circumferences into L2. Then in the
STAT menu choose CALC and the LinReg(ax +
b) function to find the equation for the regression
line.
Here, a≈3.14andb≈0,soanequationforalineof
best fit would be C = 3.14d; the equation represents
a formula for approximating the circumference of a
circle. The slope represents an approximation for pi.
44.WHICH ONE DOESN’T BELONG? Identify the
term that does not belong with the other three.
Explain your reasoning.
SOLUTION:
Circle; The other shapes are polygons.
45.CHALLENGETheverticesofarectanglewith
side lengths of 10 and 24 are on a circle of radius
13 units. Find the area between the figures.
SOLUTION:
Startbydrawingthefigure.
The shaded region of the drawing represents the
area between the figures. Next, find the area of
each figure.
Use the formula to find the area of the rectangle
(Area of a rectangle = bh).
units2
Use the formula to find the area of the circle (Area
of a circle ).
units2
Then subtract the area of the rectangle from the
area of the circle in order to find the area of the
shaded region.
Therefore, the area between the figures is about
290.93 units2.
46.REASONING Name a polygon that is always
regular and a polygon that is sometimes regular.
Explain your reasoning.
SOLUTION:
Square; by definition, all sides of a square are
congruentandallanglesmeasure90°,sotherefore
are congruent. Triangle; triangles can have all sides
and angles congruent, just two sides and angle pairs
congruent, or no sides or angles congruent.
47.OPEN ENDED Draw a pentagon. Is your pentagon
convex or concave? Is your pentagon regular or
irregular? Justify your answers.
SOLUTION:
Sample answer: The pentagon is convex, since no
points of the lines drawn on the edges are in the
interior. The pentagon is regular since all of the
angles and sides were constructed with the same
measurement, making them congruent to each other.
48.CHALLENGE A rectangular room measures 20
feet by 12.5 feet. How many 5-inch square tiles will
it take to cover the floor of this room? Explain.
SOLUTION:
Convert the dimensions from feet to inches.
Thelengthoftheroomis20×12or240inchesand
thewidthoftheroomis12.5×12or150inches.It
needs 240 5 = 48 columns of tiles and 150 5 = 30
rows of tiles to cover this space. So the number of
tilesneededis48×30or1440tiles.
49.WRITING IN MATH Describe two possible ways
that a polygon can be equiangular but not a regular
polygon.
SOLUTION:
Sample answer: If a convex polygon is equiangular
but not also equilateral, then it is not a regular
polygon. Likewise, if a polygon is equiangular and
equilateral, but not convex, then it is not a regular
polygon.
50.Find the perimeter of the figure.
A 17 cm
B 25 cm
C 28 cm
D 31 cm
SOLUTION:
The lengths of two sides are unknown.
The length of the base is 4 + 4 or 8 cm.
To find the length of the unknown vertical side,
subtract 3 from 6.
6 – 3 = 3
Add all the sides to find the perimeter.
The perimeter of the figure is 28 cm
The correct choice is C.
51.PROBABILITY In three successive rolls of a fair
number cube, Matt rolls a 6. What is the probability
of Matt rolling a 6 if the number cube is rolled a
fourth time?
F
G
H
J 1
SOLUTION:
Probability is defined as
.
The number of favorable outcomes is 1, and the total
number of outcomes is 6.
So, the probability of rolling a 6 = .
The correct choice is F.
52.SHORT RESPONSE Miguel is planning a party for
80 guests. According to the pattern in the table, how
many gallons of ice cream should Miguel buy?
SOLUTION:
From the pattern we see that for every 8 guests
Miguel needs 2 gallons of ice cream.
So,for8×10or80guestsMiguelneeds2×10or20
gallons of ice cream.
53.SAT/ACT A frame 2 inches wide surrounds a
painting that is 18 inches wide and 14 inches tall.
What is the area of the frame?
A 68 in2
B 84 in2
C 144 in2
D 252 in2
E 396 in2
SOLUTION:
With 2 inch wide frame, the dimensions of painting
with the frame becomes 22 inches by 18 inches.
Find the area of the painting with frame.
A =22·18
=396in2
Find the area of the painting with out the frame.
A =18·14
=252in2
To find the area of the frame, subtract the area of
the paint from the area of the paint with the frame.
396 – 252 = 144
The area of the frame is 144 in2.
The correct choice is C.
Determine whether each statement can be
assumed from the figure. Explain.
54. KJN is a right angle.
SOLUTION:
Yes; the symbol denotes that KJN is a right angle.
55. PLN NLM
SOLUTION:
It might appear that the angles are congruent but
there are no marks on the angles in the diagram to
indicate this. There is also nothing in the diagram that
would provide information to know that bisects
∠PLM. Because we do not know anything about the
measures of these two angles, we can not assume
that ∠PLN ∠NLM .
56. PNL and MNL are complementary.
SOLUTION:
No; we do not know whether MNP is a right
angle.
57. KLN and MLN are supplementary.
SOLUTION:
Yes; they form a linear pair.
58.TABLE TENNIS The diagram shows the angle of
play for a table tennis player. If a right-handed player
has a strong forehand, he should stand to the left of
the center line of his opponent’s angle of play.
a. What geometric term describes the center line?
b. If the angle of play shown in the diagram
measures43°,whatis mBAD?
SOLUTION:
a. The center line divides the angle into two
congruent angles. So, it is an angle bisector.
b. Since the center line is an angle bisector,
Name an appropriate method to solve each
system of equations. Then solve the system.
59.
SOLUTION:
The appropriate method to solve this system is the
elimination method, since neither of the equations
have variables with coefficients of 1 and equations
can not be simplified.
Multiply the first equation by 2 and the second
equation by 5
Substitute the value of y in one of the given
equations.
The solution of the system is: x = –3, y = –1
60.
SOLUTION:
The appropriate method to solve this system is
graphing since both equations are in slope-intercept
form and can be easily graphed.
Graph the lines on a coordinate grid.
The graphs appear to intersect at (3, –8).
So, the solution of the system of equations is x = 3, y
= –8.
61.
SOLUTION:
The appropriate method to solve this system is
substitution, since equation 1 has a variable with a
coefficient of 1.
Solve the first equation for x.
Substitute this in the second equation.
Substitute the value of y in one of the given equation.
The solution to the system is x = –4, y = –2.5
Evaluate each expression if P = 10, B = 12, h =
6, r = 3, and =5.Roundtothenearesttenth,
if necessary.
62.
SOLUTION:
Substitute.
63.
SOLUTION:
Substitute.
64.
SOLUTION:
Substitute.
65.
SOLUTION:
Substitute.
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1-6 Two-Dimensional Figures
