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6.6 Use Proportionality Theorems

6.6 Use Proportionality Theorems. Objectives. Use proportional parts of triangles Divide a segment into parts. Triangle Proportionality Theorem.

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6.6 Use Proportionality Theorems

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  1. 6.6 Use Proportionality Theorems

  2. Objectives • Use proportional parts of triangles • Divide a segment into parts

  3. Triangle Proportionality Theorem • Theorem 6.4If a line is parallel to one side of a Δ and intersects the other two sides in two distinct points, then it separates these sides into segments of proportional lengths. EG=EHGDHF *The Converse of the Δ Proportionality Theorem is also true.

  4. In and Find SU. S From the Triangle Proportionality Theorem, Example 1:

  5. Answer: Example 1: Substitute the known measures. Cross products Multiply. Divide each side by 8. Simplify.

  6. In and Find BY. B Your Turn: Answer: 15.75

  7. In and Determine whether Explain. Example 2:

  8. In order to show that we must show that Since the sides have proportional length. Answer: since the segments have proportional lengths, Example 2:

  9. In andAZ = 32. Determine whether Explain. X Answer: No; the segments are not in proportion since Your Turn:

  10. What does the Δ Proportionality Theorem Also Tell Us • The Δ Proportionality Theorem has shown us that || lines cut the sides of a Δ into proportional parts. Three or more parallel lines also separate transversals into proportional parts.

  11. Additional Theorems • Theorem 6.6 If 3 or more || lines intersect 2 transversals, then they divide the transversals proportionally. • AB = DE BC EF A D E B C F

  12. Additional Theorems • Theorem 6.7 If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. • PT = TS PU US T P U S

  13. Example 3: In the figure, Larch, Maple, and Nuthatch Streets are all parallel. The figure shows the distances in city blocks that the streets are apart. Find x.

  14. Example 3: Notice that the streets form a triangle that is cut by parallel lines. So you can use the Triangle Proportionality Theorem. Triangle Proportionality Theorem Cross products Multiply. Divide each side by 13. Answer: 32

  15. Your Turn: In the figure, Davis, Broad, and Main Streets are all parallel. The figure shows the distances in city blocks that the streets are apart. Find x. Answer: 5

  16. Example 4: Find x and y. To find x: Given Subtract 2x from each side. Add 4 to each side.

  17. The segments with lengths are congruent since parallel lines that cut off congruent segments on one transversal cut off congruent segments on every transversal. Example 4: To find y:

  18. Example 4: Equal lengths Multiply each side by 3 to eliminate the denominator. Subtract 8y from each side. Divide each side by 7. Answer: x = 6; y = 3

  19. Your Turn: Find a and b. Answer: a = 11; b = 1.5

  20. Assignment • Geometry Workbook Pgs. 118 – 119 #1 – 25

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