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7.5

7.5. ESSENTIAL QUESTION: How do you use the Triangle Proportionality Theorem and its C onverse in solving missing parts?. Theorem 7.4 Triangle Proportionality Theorem.

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7.5

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  1. 7.5 ESSENTIAL QUESTION: • How do you use the Triangle Proportionality Theorem and its Converse in solving missing parts?

  2. 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.

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

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

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

  7. Checkpoint Find Segment Lengths and Determine Parallels 10 ANSWER 2.

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

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

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

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

  12. 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. QS = PT = (10) = 5 1 1 2 2 The length of QS is 5. ANSWER

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

  14. Homework • Worksheet 7.5A

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