Bell Ringer

Proportions and Similar Triangles

Example 1 Find Segment Lengths Find the value of x. 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 = 8x8x 8 Divide each side by 8. SOLUTION CD DB = CE EA Triangle Proportionality Theorem 6 = x Simplify.

Example 2 Find Segment Lengths Find the value of y. 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 PQ QR = PT TS Triangle Proportionality Theorem SOLUTION You know that PS = 20 and PT = y. By the Segment Addition Postulate, TS = 20 – y.

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

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

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

Checkpoint Find Segment Lengths and Determine Parallels Given the diagram, determine whether QR is parallel to ST. Explain. ANSWER Converse of the Triangle Proportionality Theorem. = Yes; || so QR ST by the ≠ no; ANSWER Now You Try

A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle.

Example 4 Use the Midsegment Theorem Find the length of QS. 1 2 QS = PT = (10) = ANSWER The length of QS is 5. 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.

Checkpoint Use the Midsegment Theorem Find the value of the variable. ANSWER ANSWER Use the Midsegment Theorem to find the perimeter of  ABC. Now You Try

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