Bell Ringer

Triangle Mid-segments

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

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

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

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

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

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

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

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

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

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Page 390

Complete Pages 390-392 #s 10-28 even only