Geometry unit 7.4

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  1. 1. 7 UNIT 7..44 SSIIMMIILLAARRIITTYY IINN RRIIGGHHTTHolt GeometryTTRRIIAANNGGLLEESS
  2. 2. Warm Up1. Write a similarity statementcomparing the two triangles.ADB ~ EDCSimplify.2. 3.Solve each equation.4. 5. 2x2 = 505
  3. 3. ObjectivesUse geometric mean to find segmentlengths in right triangles.Apply similarity relationships in righttriangles to solve problems.
  4. 4. Vocabularygeometric mean
  5. 5. In a right triangle, an altitude drawn from thevertex of the right angle to the hypotenuse formstwo right triangles.
  6. 6. 7.3
  7. 7. Example 1: Identifying Similar Right TrianglesWrite a similaritystatement comparing thethree triangles.Sketch the three right triangles with theangles of the triangles in correspondingpositions.WZBy Theorem 8-1-1, UVW ~ UWZ ~ WVZ.
  8. 8. Check It Out! Example 1Write a similarity statementcomparing the three triangles.Sketch the three right triangles withthe angles of the triangles incorresponding positions.By Theorem 8-1-1, LJK ~ JMK ~ LMJ.
  9. 9. Consider the proportion . In this case, themeans of the proportion are the same number, andthat number is the geometric mean of the extremes.The geometric mean of two positive numbers is thepositive square root of their product. So the geometricmean of a and b is the positive number x suchthat , or x2 = ab.
  10. 10. Example 2A: Finding Geometric MeansFind the geometric mean of each pair ofnumbers. If necessary, give the answer insimplest radical form.4 and 25Let x be the geometric mean.x2 = (4)(25) = 100 Def. of geometric meanx = 10 Find the positive square root.
  11. 11. Example 2B: Finding Geometric MeansFind the geometric mean of each pair ofnumbers. If necessary, give the answer insimplest radical form.5 and 30Let x be the geometric mean.x2 = (5)(30) = 150 Def. of geometric meanFind the positive square root.
  12. 12. Check It Out! Example 2aFind the geometric mean of each pair ofnumbers. If necessary, give the answer insimplest radical form.2 and 8Let x be the geometric mean.x2 = (2)(8) = 16 Def. of geometric meanx = 4 Find the positive square root.
  13. 13. Check It Out! Example 2bFind the geometric mean of each pair ofnumbers. If necessary, give the answer insimplest radical form.10 and 30Let x be the geometric mean.x2 = (10)(30) = 300 Def. of geometric meanFind the positive square root.
  14. 14. Check It Out! Example 2cFind the geometric mean of each pair ofnumbers. If necessary, give the answer insimplest radical form.8 and 9Let x be the geometric mean.x2 = (8)(9) = 72 Def. of geometric meanFind the positive square root.
  15. 15. You can use Theorem 8-1-1 to write proportionscomparing the side lengths of the triangles formedby the altitude to the hypotenuse of a right triangle.All the relationships in red involve geometric means.
  16. 16. 7.3.17.3.2
  17. 17. Example 3: Finding Side Lengths in Right TrianglesFind x, y, and z.62 = (9)(x) 6 is the geometric mean of9 and x.x = 4 Divide both sides by 9.y2 = (4)(13) = 52 y is the geometric mean of4 and 13.Find the positive square root.z2 = (9)(13) = 117 z is the geometric mean of9 and 13.Find the positive square root.
  18. 18. Helpful HintOnce youve found the unknown side lengths,you can use the Pythagorean Theorem to checkyour answers.
  19. 19. Check It Out! Example 3Find u, v, and w.92 = (3)(u) 9 is the geometric mean ofu and 3.u = 27 Divide both sides by 3.w2 = (27 + 3)(27) w is the geometric mean ofu + 3 and 27.Find the positive square root.v2 = (27 + 3)(3) v is the geometric meanofFind the positivue + s 3q uaanrde 3ro. ot.
  20. 20. Example 4: Measurement ApplicationTo estimate the height of aDouglas fir, Jan positionsherself so that her lines ofsight to the top and bottomof the tree form a 90angle. Her eyes are about1.6 m above the ground,and she is standing 7.8 mfrom the tree. What is theheight of the tree to thenearest meter?
  21. 21. Example 4 ContinuedLet x be the height of the tree above eye level.(7.8)2 = 1.6xx = 38.025 387.8 is the geometric mean of1.6 and x.Solve for x and round.The tree is about 38 + 1.6 = 39.6, or 40 m tall.
  22. 22. Check It Out! Example 4A surveyor positions himselfso that his line of sight tothe top of a cliff and his lineof sight to the bottom forma right angle as shown.What is the height of thecliff to the nearest foot?
  23. 23. Check It Out! Example 4 ContinuedLet x be the height of cliff above eye level.(28)2 = 5.5x 28 is the geometric mean of5.5 and x.Divide x 142.5 both sides by 5.5.The cliff is about 142.5 + 5.5, or148 ft high.
  24. 24. Lesson Quiz: Part IFind the geometric mean of each pair ofnumbers. If necessary, give the answer insimplest radical form.1. 8 and 18122. 6 and 15
  25. 25. Lesson Quiz: Part IIFor Items 36, use RST.3. Write a similarity statement comparing thethree triangles.RST ~ RPS ~ SPT44. If PS = 6 and PT = 9, find PR.5. If TP = 24 and PR = 6, find RS.6. Complete the equation (ST)2 = (TP + PR)(?).TP
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