Midsegments of Triangles: Properties and Applications, Slides of Reasoning

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Warm Up #

**1. Use Representations to Communicate Mathematical Ideas (1)E

2. Find the midpoint of** 𝑹𝑺 is where the endpoints are R (‒8, 9) and S (0, ‒7). What are the coordinates of R****?

N (‒4, 1) Find the distance between two cities: Augusta and Brookline. 𝒅 = (𝒙 𝟐

𝟏

𝟐

• (𝒚 𝟐

𝟏

𝟐

• A frame house
• Ladder
• Bridge
• Porch swing
• Ferris wheel
• Architecture

All supported additionally by the midsegment bars.

SWBAT

 Solve for side lengths and angle measures in

similar triangles with midsegments.

By the end of today’s lesson, Connect to Mathematical Ideas (1)(F) Why it matters in LIFE : Midsegments support many structures in our world like the floors of a house, ladder, or bridge. Why it matters in THIS CLASS : Sides and angles are the two fundamentals that make up triangles. Midsegments allow us to solve for both missing sides and angles in triangles.

4.Label the midpoints L and N and draw segment LN 5.Fold the triangle on LN and label D as shown. 6.Fold A to D and Fold B to D****. L (^) N Label the vertices M and P as shown.

• What is the relationship between MP and AB****? How do you know?
• What conjecture can you make about the relationship between LN and AB****?

Connect to Mathematical Ideas (1)(F)

Prior Knowledge: Communicate Mathematical Ideas (1)(G) The coordinate plan at the right is the map showing the locations of your home at the origin, the school at S (‒10, 0), the library is at L (‒5,0), Shell Gas Station at F (‒2, ‒4), and the grocery is at G (‒4, ‒8). A. What do you notice about the location of the library? S (‒10, 0) L (‒5, 0)^ H (0, 0) F (‒2, ‒4) G (‒4, ‒8) B. What do you notice about the location of Shell Gas Station?

• Half the distance of 𝑆𝐻
• L (–5, 0) is the midpoint of 𝑆𝐻
  • F (–2, – 4) is the midpoint of 𝐺𝐻
  • Half the distance of 𝐺𝐻 The units of the map are in miles.

F (‒2, ‒4) G (‒4, ‒8) Your parent decided to pick up some groceries before returning to the school to pick up your older sibling, that is participating in after school math tutoring program, at S (‒10, 0). How far is the grocery store to the school? The units of the map are in miles.

2

1

2

• (𝑦 2

1

2 𝑑(𝑆𝐺) = (− 4 − (− 10 )) 2

• (− 8 − 0 ) 2 = ( 6 ) 2
• (− 8 ) 2 = 36 + 64 = 100 ⇒^10 miles S (‒10, 0) L (‒5, 0)^ H (0, 0) Prior Knowledge: What if?

Connecting and Analyze Mathematical Relationships (1)(F) F (‒2, ‒4) G (‒4, ‒8)

1. What is the conjecture for point L?
2. What is the conjecture for point F?

3. Compare segments 𝑳𝑭 and 𝑺𝑮.

4. Justify if 𝑳𝑭 ∥ 𝑺𝑮.

The units of the map are in miles. S (‒10, 0) L (‒5, 0)^ H (0, 0)

L is the midpoint of 𝑆𝐻

F is the midpoint of 𝐺𝐻

1 2

Slope: 𝑚 = 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 𝑚 𝐿𝐹 = − 4 − 0 − 2 −(− 5 ) = − 4 3 = − 4 3 𝑚 𝑆𝐺 = − 8 − 0 − 4 −(− 10 ) = − 8 6 = − 4 3

The triangular face of the Rock and Roll Hall of Fame in Cleveland, Ohio, is isosceles. The length of the base is 229 ft. 6 in. Each leg is divided into four congruent parts by the red segments. What is the length of the yellow segment? Explain your reasoning. 229 ft. 6 in Example 2: Explain Mathematical Ideas (1)(G)

It is given that the red segments divide the legs into four congruent parts, the yellow segment is a midsegment of the triangular face of the building, so its length is one haft the length of the base; therefore, the white segment is 114 ft. and 9 in. 229 ft. 6 in Example 2: Explain Mathematical Ideas (1)(G)

Example 4: Finding the value of x.

8 =

5𝑥 − 4 ∆^ Midsegment^ Theorem. 8 =^ Distributive Property

𝑥 − 2 10 =^ Combined Like Terms

𝑥 20 = 5 𝑥 Multiply both sides by 2 4 = 𝑥^ Divide both sides by 5

Example 5: If EC = 3 x – 2 and AD = 2 x + 8. Find EC.

EC =

𝐴𝐷 ∆^ Midsegment^ Theorem

(2𝑥 + 8 ) Substitution

3 𝑥 – 2= 𝑥 + 4 Distributive Property

2 𝑥 = 6 Combined Liked Terms

Divide both sides by 2. EC = 3(3) – 2 ⇔ 7