1. Bell Ringer

2. Triangle Mid-segments

3. Midsegment of a triangle
A Midsegment of a triangle is the segments that connects the midpoints to two sides of a triangle.

4. Triangle Proportionality Theorem
Example 1
Find Segment Lengths
Find the value of x.
SOLUTION CD DB = CE EA
Triangle Proportionality Theorem 4 8 x 12 =
Substitute 4 for CD, 8 for DB, x for CE, and 12 for EA.
4 · 12 = 8 · x
Cross product property
48 = 8x
Multiply. 48 8 = 8x
Divide each side by 8.
6 = x
Simplify. 4

5. Triangle Proportionality Theorem
Example 2
Find Segment Lengths
Find the value of y.
SOLUTION
You know that PS = 20 and PT = y. By the Segment Addition Postulate, TS = 20 – y. PQ QR = PT TS
Triangle Proportionality Theorem 3 9 y
20 – y =
Substitute 3 for PQ, 9 for QR, y for PT, and (20 – y) for TS.
3(20 – y) = 9 · y
Cross product property
60 – 3y = 9y
Distributive property 5

6. 60 – 3y + 3y = 9y + 3y Add 3y to each side. 60 = 12y Simplify. 60 12y
Example 2
Find Segment Lengths
60 – 3y + 3y = 9y + 3y
Add 3y to each side.
60 = 12y
Simplify. 60 12 =
12y
Divide each side by 12.
5 = y
Simplify. 6

7. Find and simplify the ratios of the two sides divided by MN.
Example 3
Given the diagram, determine whether MN is parallel to GH.
Determine Parallels
SOLUTION
Find and simplify the ratios of the two sides divided by MN. LM MG = 56 21 8 3 LN NH 48 16 1
ANSWER
Because ≠ 3 1 8
, MN is not parallel to GH. 7

8. Now You Try  ANSWER 8 ANSWER 10
Find Segment Lengths and Determine Parallels
Find the value of the variable. 1.
ANSWER 8 2.
ANSWER 10

9. Now You Try  ≠ 17 23 15 21 no; ANSWER ANSWER
Find Segment Lengths and Determine Parallels
Now You Try  3.
Given the diagram, determine whether QR is parallel to ST. Explain. ≠ 17 23 15 21
no;
ANSWER 4.
ANSWER
Converse of the Triangle Proportionality Theorem. = 6 12 4 8
Yes; ||
so QR ST by the

10. 1 QS = PT = (10) = 5 2 ANSWER The length of QS is 5.
Example 4
Use the Midsegment Theorem
Find the length of QS.
SOLUTION
From the marks on the diagram, you know S is the midpoint of RT, and Q is the midpoint of RP. Therefore, QS is a midsegment of PRT. Use the Midsegment Theorem to write the following equation. 1 2
QS = PT = (10) = 5
ANSWER
The length of QS is 5. 10

11. Now You Try  ANSWER 8 ANSWER 28 ANSWER 24 Use the Midsegment Theorem
Find the value of the variable. 5.
ANSWER 8 6.
ANSWER 28 7.
Use the Midsegment Theorem to find the perimeter of ABC.
ANSWER 24

12. Now You Try 

13. Now You Try 

14. Now You Try 

15. Now You Try 

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17. Complete Pages
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