7.5 ESSENTIAL QUESTION: How do you use the Triangle Proportionality Theorem and its Converse in solving missing parts?

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Presentation transcript:

7.5 ESSENTIAL QUESTION: How do you use the Triangle Proportionality Theorem and its Converse in solving missing parts?

Theorem 7.4 Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

Example 1 Find the value of x. SOLUTION CD DB = CE EA 4 8 x 12 = 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

Theorem 7.5 Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Given the diagram, determine whether MN is parallel to GH. 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 = 56 21 8 3 LN NH 48 16 1 ANSWER Because ≠ 3 1 8 , MN is not parallel to GH. 7

Find the value of the variable. Checkpoint Find Segment Lengths and Determine Parallels Find the value of the variable. 1. ANSWER 8

Checkpoint Find Segment Lengths and Determine Parallels 2. ANSWER 10

Given the diagram, determine whether QR is parallel to ST. Explain. Checkpoint Find Segment Lengths and Determine Parallels 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

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

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

The Midsegment Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long.

Example 4 Find the length of QS. SOLUTION 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. 15

Find the value of the variable. Checkpoint Use the Midsegment Theorem Find the value of the variable. 5. ANSWER 8 6. ANSWER 28

Homework Worksheet 7.5A