Concept Byte: Investigating Midsegments

Use With Lesson 5-1

TECHNOLOGY

Activity

• Step 1 Use geometry software to draw and label Δ A B C . cap delta eh b c .  Construct the midpoints D and E of A B ¯ eh b bar  and A C ¯ . eh c bar , .  Connect the midpoints with a midsegment.
• Step 2 Measure D E ¯ d e bar  and B C ¯ . b c bar , .  Calculate D E B C . fraction d e , over b c end fraction . .
• Step 3 Measure the slopes of D E ¯ d e bar  and B C ¯ . b c bar , .
• Step 4 Manipulate the triangle and observe the lengths and slopes of D E ¯ d e bar  and B C ¯ . b c bar , .

Exercises

1. Make a Conjecture Make conjectures about the lengths and slopes of midsegments.
2. Construct the midpoint F of B C ¯ . b c bar , .  Then construct the other two midsegments of Δ A B C . cap delta eh b c .  Test whether these midsegments support your conjectures in Exercise 1.
3. Δ A B C cap delta eh b c  and the three midsegments form four small triangles.
   1. Measure the sides of the four small triangles and list those that you find are congruent.
   2. Make a Conjecture Make a conjecture about the four small triangles formed by a triangle and its three midsegments.

For the remaining exercises, assume your conjectures in Exercises 1 and 3 are true.

4. What can you say about the areas of the four small triangles in the window above?
5. 1. How does Δ A B C cap delta eh b c  compare to each small triangle in area?
   2. How does Δ A B C cap delta eh b c  compare to each small triangle in perimeter?
6. Construct the three midsegments of Δ D E F . cap delta d e f .  Label this triangle Δ G H I . cap delta g h i .
   1. How does Δ A B C cap delta eh b c  compare to Δ G H I cap delta g h i  in area?
   2. How does Δ A B C cap delta eh b c  compare to Δ G H I cap delta g h i  in perimeter?
   3. Suppose you construct the midsegment triangle inside Δ G H I . cap delta g h i .  How would Δ A B C cap delta eh b c  compare to this third midsegment triangle in area and perimeter?

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