GRE Geometry: A Comprehensive Guide to Mastering Shapes and Formulas, Study Guides, Projects, Research of Mathematics

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Chapter 8

Geometry

ANGLING FOR A BETTER GRADE?

Now that you’ve reached this chapter, you may be having a flashback to your freshman year in high school when you first came in contact with theorems, postulates, and definitions, all woven together to form the geometric proof. Well, relax. GRE geometry questions have little to do with deductive reasoning. You’re much more likely to be tested on the basic formulas involving area, perimeter, volume, and angle measurements. As you work through the GRE Math, you’ll find that there is a basic battery of terms and formulas that you should know for the geometry questions that do come your way. Before we get to those, let’s look at some techniques.

• Plug In. If a problem tells you that a rectangle is x inches long and y inches wide, plug in some real numbers to help the question take on a more tangible quality. (And if there is more than one variable, remember to plug in a different number for each one.)
• Use Ballparking. If a diagram is drawn to scale, you can sometimes estimate the right answer and eliminate all the answer choices that don’t come close.
• Redraw to Scale. If a diagram is not drawn to scale—and for problem solving questions you’ll know because you’ll see the words “Note: Figure not drawn to scale” right below the picture—redraw it to make it look like it’s supposed to look like. Drawings like this are meant to confuse you by suggesting that the figure looks how it’s represented in the problem. Redrawing the figure to scale helps you avoid falling into that trap.

Drawn to Scale: Problem Solving Problem Solving figures are typically drawn to scale. When they are not drawn to scale, ETS adds “Note: Figure not drawn to scale” beneath the figure.

Drawn to Scale: Quant Comp Quant Comp figures are often drawn to scale, but sometimes they aren’t to scale. If they aren’t, ETS does not add any sort of warning like they do for problem solving. Check the information in the problem carefully and be suspicious of the figure.

BASIC HINTS FOR GEOMETRY QUESTIONS

Geometry is a special science all its own, but that doesn’t mean it marches to the beat of an entirely different drummer. Many of the techniques you’ve learned for other problems will work here as well.

So let’s start with our pal Euclid and his three primary building blocks of measured space: points, lines, and planes.

LINES AND ANGLES

Two points determine a line, and two intersecting lines form an angle measured in degrees. There are 360° in a complete circle, so halfway around the circle forms a straight angle, which measures 180°, and half of that is a right

l 1 || l 2

Notice that for the figure above, the acute (small) angles labeled 1, 3, 5, and 7 are all the same, because l 1 is parallel

with l 2. You also know that the angles labeled 2, 4, 6, and 8 are all the same for the same reason. Whenever the GRE

states that two lines are parallel, look to see if the question is actually testing this concept.

TRIANGLES

Three points determine a triangle, and all triangles have three sides and three angles. The sum of the measures of the angles inside a triangle is 180°. The sides and angles are related. Just remember that the longest side is always opposite the largest angle, and the shortest side is always opposite the smallest angle.

Types of Triangles

The properties of triangles start to get more interesting when some or all of the sides have the same length.

• Isosceles Triangles : If two sides of a triangle have the same length, the triangle is isosceles. The relationship between sides and angles still goes; if two sides of a triangle are the same length, then the angles opposite those sides have the same degree measure.
• Equilateral Triangles: Equilateral triangles have three equal sides and three equal angles. Because the

sum of the angle measures is 180°, each angle in an equilateral triangle measures , or 60°.

Right Triangles

Right triangles contain exactly one right angle and two acute angles. The perpendicular sides are called legs, and the longest side (which is opposite the right angle) is called the hypotenuse.

The Pythagorean Theorem

Whenever you know the length of two sides of a right triangle, you can find the length of the third side by using the Pythagorean Theorem.

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Here’s How to Crack It

The figure consists of two special triangles; consider the 45 : 45 : 90 triangle on the left. If PN = 4, then MN = 4 and

MP = 4. The height PN also helps you find the lengths of the 30 : 60 : 90 triangle on the right. Because PN is the

short side, the hypotenuse PO is twice as long, or 8 inches long. The other side, NO , measures 4. The perimeter of

triangle MPO is therefore 4 + 4 + 8 + 4 , so the answer is (C).

Area

The formula for the area of a triangle is A = bh , where b is the length of the base and h is the perpendicular distance

from the vertex to the base (also known as the height, or the altitude). Because the legs of a right triangle are

perpendicular, you can use the length of one leg as the base and the length of the other as the height.

In the diagram above, each of the triangles has the same base and a height of the same measure. Therefore, each triangle has the same area.

How Long Is the Third Side?

If you know the lengths of two sides of a triangle, you can use a simple formula to determine how long and how short the third side could possibly be.

If the lengths of two sides of a triangle are x and y , respectively, the length of the third side must be less than x + y and greater than | x – y |.

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Here’s How to Crack It This is a “third-side” problem disguised as a word problem about three towns. Because the towns do not lie along a straight line, they form a triangle; one side is 65 miles long, and the other is 40 miles long. Therefore, the third side (the length between towns A and C) must be greater than 65 − 40, or 25, and less than 65 + 40, or 105. (Remember, the distance has to be greater than 25, so it can’t be equal to 25, nor can it be equal to 105.) Therefore, the correct answer is (C).

Triangle Quick Quiz

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Explanations for Triangle Quick Quiz

1. The first two measures are 50° and 65°, so their total measure is 115°. The third angle must measure 180 − 115, or 65 °. The answer is (C).
2. Since the triangle is isosceles, then the third side must be either 11.5 or 13.7. If the third side is 11.5, then the perimeter is 36.7. If the third side is 13.7, then the perimeter is 38.9. The answer is (D).
3. The height is 5 cm so the base is 10 cm. The area is therefore (5)(10), or 25 cm^2. The answer is (D).
4. Using the Pythagorean Theorem, you can find the third side. 25^2 − 15^2 = 400, and the square root of 400 is
   20. The perimeter equals 15 + 20 + 25, or 60. You can also find the third side based on the 3 : 4 : 5 Pythagorean triple (just multiply the whole ratio by 5). The answer is 60.
5. Draw out the right triangle described in the question. By traveling due south and then due east, you get a right triangle with legs of length 5 and 12. Now you can solve for the length of the hypotenuse, or the distance from the store to the house. It equals 13 from the 5 : 12 : 13 Pythagorean triple. The question asks how many fewer miles Mike would travel if he could travel in a straight line. Mike travels a total of 5 + 12 = 17 miles as opposed to 13 miles if he could travel in a straight line. 17 − 13 = 4, so he would save 4 miles, which is (B).
6. The only single-digit prime numbers are 2, 3, 5, and 7; remember, though, that triangles must conform to the Third Side Rule, which states that the largest side of a triangle must be less than the sum of the other two sides. Only (E) is the sum of 3 sides that meet both requirements: 3, 5, and 7. Choices (B), (C), and (D) can also be reached by adding 3 distinct single digit primes—2, 3, and 5; 2, 3, and 7; and 2, 5, and 7, respectively—but all violate the Third Side Rule. If you selected (A) or (F), remember to use only distinct values for the sides. The answer is (E).

QUADRILATERALS

A quadrilateral is any figure that has four sides, and the same types of quadrilaterals—parallelograms, rectangles, and squares—show up over and over again on the GRE. Regardless of their shape or size, however, one thing is true of all four-sided figures: They can be divided into two triangles. From this, you can determine a couple of things:

Degrees : Because every quadrilateral can be divided into 2 triangles, all quadrilaterals obey what we call the Rule of 360: There are 180° in a triangle, so there are 2 × 180, or 360, degrees in every quadrilateral.

Area : The area of a triangle is bh , so the area of a parallelogram is 2 × bh , or bh.

Properties, Area, and Perimeter

Quadrilaterals are often referred to as a “family” because they share lots of characteristics. For example, every rectangle is a parallelogram, so rectangles have every characteristic that a parallelogram has, and they also happen to have four right angles. Ditto for a square, which is just a rectangle that has four equal sides.

Here is a handy chart to help you keep track of all the various properties and the formulas for area and perimeter.

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Explanations for Quadrilateral Quick Quiz

1. The sum of the given angles is 70° + 130°, or 200°, so the sum of the remaining angles must be 360° −

200°, or 160°. These angles are the same size, so they must each measure , or 80°. The answer is (B).

2. If the width of the garden is w , then the length is five times that, or 5 w. Plug these into the formula for perimeter (2 l + 2 w ) and solve: 2(5 w ) + 2 w = 36, so 12 w = 36 and w = 3. The garden is 3 feet wide. The answer is (A).
3. The area of a parallelogram is base × height, so the parallelogram’s area is 10 × 15, or 150 cm^2. The answer is (D).

CIRCLES

A circle represents all the points that are a fixed distance away from a certain point (called the center). The fixed distance from the center to the edge is the radius, and all radii are equal in length. When a radius is rotated 360° around the center, the circumference (the perimeter of the circle) is formed; any segment connecting two points on the circumference is called a chord. The diameter is the longest chord that can be drawn on a circle; it goes through the center and is twice as long as the radius.

Area and Circumference

Circles are wondrous things, because they gave us π. One day, a Greek mathematician with a lot of time on his hands began measuring circumferences ( C ) of circles and dividing those distances by the diameters ( d ), and he kept getting the same number: 3.141592….He thought this was pretty cool, but also hard to remember, so he renamed it “ p .” He was Greek, though, so he used the Greek letter p , which is π.

From this discovery we find that = π, and this can be rewritten as C = π d. This is the most common formula for

finding the circumference of a circle. A diameter is twice as long as a radius ( d = 2 r ), so you can also write the

formula as C = 2π r. The formula for the area of a circle is A = π r^2.

• Area of a circle = π r^2
• Circumference of a circle = 2π r

Notice that the radius, r , is in both of those formulas? The radius is the most important part of a circle to know. Once you know the radius, you can easily find the diameter, the circumference, or the area. So if you’re ever stuck on a circle question, find the radius.

One quick note about π. Although it is true that π ≈ 3.1415 (and so on and so on), you won’t have to use that too often on the GRE. Don’t worry about memorizing π beyond the hundredths digit: It’s 3.14. Even that is more precise than the GRE typically requires. Most answers are going to be in terms of π, which means that the GRE is much more likely to have 5π as an answer choice than it is to have 15.707. So don’t multiply out π unless you absolutely have to. Most of the time, each individual π will either cancel out or be in the answer choices.

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Here’s How to Crack It

There are 360° in a circle and 12 numbers on the face of a clock. Therefore, the measure of the central angle between

each numeral on the clock (say, between the 12 and the 1) is , or 30°. There are four such central angles between

the 9 and the 1, so the central angle is 4 × 30, or 120°. The radius of the circle is 9 inches, so the area of the whole

clock is π(9)^2 , or 81π. To find the area of the sector, use the formula: 81π × = 27π. The correct answer is (C).

Circle Quick Quiz

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Explanations for Circle Quick Quiz

1. Because C = π d , the diameter of a circle with a circumference of 6π inches is 6 inches and thus the radius is 3 inches. The answer is (B).
2. The diameter of the circle is 6, so the radius is half that, or 3. The area of the circle is π(3)^2 , or 9π square inches. The answer is (C).
3. The area of the pie is π r^2 , or 64π, so each slice has an area of square inches. This converts to.

The answer is (C).

4. There are 360° in a circle, so the measure of each central angle is , or 60°. The answer is (C). (Notice

that you did not have to use the radius measurement to answer this question.)

THE COORDINATE PLANE

When you plot points on the Cartesian plane, you’ll most likely be asked to (1) find the distance between two of them, (2) find the slope of the line that connects the two points, or (3) find the equation of the line they define.

Distance in the Coordinate Plane

If you need to find the distance between two points in the coordinate plane, draw a right triangle. The hypotenuse of the triangle is the distance between the two points, and the legs are the differences in the x- and y -coordinates of the two points.

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Here’s How to Crack It Start by drawing a simple coordinate plane on your scratch paper. You don’t need to mark out each and every tick mark; this is just to get a rough idea of where the points are. Your drawing will probably look something like this:

Now find the length of each leg. The bottom leg is the distance between the two x -coordinates. From −3 to 2 is a total of 5 units (or, |−3 −2| = 5). The height is the distance between the two y -coordinates. From 3 to 15 is 12 units (or, |− − 15| = 12). So you have a triangle with sides of length 5 and 12. You can either use the Pythagorean Theorem to find the hypotenuse, or use the fact that you have a 5 : 12 : 13 triangle (one of the Pythagorean triples), which means that the line is 13 units long.

Many two-dimensional distance problems are really Pythagorean Theorem problems in disguise. The GRE will often hide this using questions in which people travel north/south and east/west.