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5-Minute Check on Lesson 6-4

Transparency 6-5. 5-Minute Check on Lesson 6-4. R. Refer to the figure If QT = 5, TR = 4, and US = 6, find QU. If TQ = x + 1, TR = x – 1, QU = 10 and QS = 15, find x . Refer to the figure If AB = 5, ED = 8, BC = 11, and DC = x – 2, find x so that BD // AE.

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5-Minute Check on Lesson 6-4

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  1. Transparency 6-5 5-Minute Check on Lesson 6-4 R • Refer to the figure • If QT = 5, TR = 4, and US = 6, find QU. • If TQ = x + 1, TR = x – 1, QU = 10 and QS = 15, find x. • Refer to the figure • If AB = 5, ED = 8, BC = 11, and DC = x – 2, find x so that BD // AE. • If AB = 4, BC = 7, ED = 5, and EC = 13.75, determine whetherBD // AE. T 7.5 S Q U 3 B A C D E 19.6 Yes Click the mouse button or press the Space Bar to display the answers.

  2. Lesson 6-5 Parts of Similar Triangles

  3. Objectives • Recognize and use proportional relationships of corresponding perimeters of similar triangles • Recognize and use proportional relationships of corresponding angle bisectors, altitudes, and medians of similar triangles

  4. Vocabulary • None New

  5. Theorems • If two triangles are similar then • The perimeters are proportional to the measures of corresponding sides • The measures of the corresponding altitudes are proportional to the measures of the corresponding sides • The measures of the corresponding angle bisectors of the triangles are proportional to the measures of the corresponding sides • The measures of the corresponding medians are proportional to the measures of the corresponding sides • Theorem 6.11: Angle Bisector Theorem: An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides

  6. Special Segments of Similar Triangles P If ∆PMN ~ ∆PRQ, then PM PN MN special segment ----- = ----- = ----- = ------------------------- AB AC BC special segment ratios of corresponding special segments = scaling factor (just like the sides) in similar triangles Example: PM 1 median PQ 1 ----- = --- ----------------- = --- AB 3 median AD 3 Special segments are altitudes, medians, angle and perpendicular bisectors M N Q A C B D

  7. Similar Triangles -- Perimeters If ∆PMN ~ ∆PRQ, then Perimeter of ∆PMN PM PN MN ------------------------- = ----- = ----- = ----- Perimeter of ∆PRQ PR PQ RQ ratios of perimeters = scaling factor (just like the sides) P M N Q R

  8. Angle Bisector Theorem - Ratios P If PN is an angle bisector of P, then the ratio of the divided opposite side, RQ, is the same as the ratio of the sides of P, PR and PQ PR RN ----- = ----- PQ NQ R N Q

  9. K Example 1 In the figure, ∆EFG~ ∆JKL, ED is an altitude of ∆EFGand JI is an altitude of ∆JKL. Find x if EF=36, ED=18, and JK=56. Write a proportion. Cross products Divide each side by 36. Answer: Thus, JI = 28.

  10. N Example 2 In the figure, ∆ABD ~ ∆MNP and AC is an altitude of ∆ABD and MO is an altitude of ∆MNP. Find x if AC=5, AB=7 and MO=12.5 Answer: 17.5

  11. are medians of since and If two triangles are similar, then the measures of the corresponding medians are proportional to the measures of the corresponding sides. This leads to the proportion Example 3 The drawing below illustrates two poles supported by wires with ∆ABC~∆GED , AFCF, and FGGC DC. Find the height of the pole EC.

  12. measures 40 ft. Also, since both measure 20 ft. Therefore, Example 3 cont Write a proportion. Cross products Simplify. Divide each side by 80. Answer: The height of the pole is 15 feet.

  13. Summary & Homework • Summary: • Similar triangles have perimeters proportional to the corresponding sides • Corresponding angle bisectors, medians, and altitudes of similar triangles have lengths in the same ratio as corresponding sides • Homework: • Page 321 (10, 22-26) MUST SHOW WORK!

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