Unit 2: World of Polygons Part 1
Lesson 8
Student Activity
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What can you discover about the segment joining midpoints of two sides of a triangle?
Opening Activity
You will be given three triangles. Do the following:
• Measure the length of each side of the three triangles.
• Find the midpoint of each side of the three triangles. Use a ruler or a compass. Mark the midpoints.
• In each triangle connect the midpoints of two sides. You should have created three segments inside
each triangle.
• Measure the lengths of each segment inside the triangles.
Now you are going to observe the lines and the data you collected. Think about the following two
questions. Be prepared to discuss your findings with the rest of the class.
•
What do you observe about the relationship between the segments formed by joining the midpoints
and the sides of the triangles?
•
What do you observe about the lengths of the segments formed by joining the midpoints and
the lengths of the sides of the triangles?
Activity Two
What did we just observe in the first activity?
Theorem: The line segment connecting the midpoints of two sides of a triangle is parallel to the
third side and is congruent to one half of the third side.
Read the following proof by construction then see if can support the claim.
Consider the triangle ABC with the midpoint of AB labeled M.
Now construct a line through M parallel to BC. Label the point of intersection
of this line with BC by P.
Now construct a line through M parallel to AC. Label the point
of intersection of this line with AC by Q
Claim: AMP is congruent to MBQ. Why?
(How does this help us prove the midpoint theorem?)
Activity Three
Given: D, E and F are midpoints
1) Given the triangle and information above describe all the things you know and the reason that
supports it. This should be information about sides and angles.
2) If m<DBF = 47o and m<FCE = 68o then what is m<DAE? Explain your thinking.
3) If the perimeter of triangle ABC is 42 units, segment DE = 8 units and segment AB = 14 units find the
length of segment DF.
4) Revisit your data from the earlier investigation. Observe the perimeters for both the given triangle and
the triangle formed by joining the midpoints. What observations can you make? Will this always be
true?