Geom 9point1

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Geom 9point1

  1. 1. Similar Right Triangles <ul><li>Objectives: </li></ul><ul><li>Solve problems involving similar right triangles formed by the altitude drawn to the hypotenuse of a right triangle </li></ul><ul><li>Solve problems using geometric means </li></ul>
  2. 2. Make some similar right triangles <ul><li>Cut an index card along the diagonal </li></ul><ul><li>On one of the right triangles, draw an altitude from the right angle to the hypotenuse. (It must make a right angle with the hypotenuse.) </li></ul><ul><li>You should now have 3 right triangles. Are they similar? </li></ul>
  3. 3. Theorem <ul><li>If the altitude is drawn to the hypotenuse of a right triangle, then the 2 triangles formed are similar to the original triangle and to each other. </li></ul>
  4. 4. Find the height of the roof
  5. 5. Find the height of the roof <ul><li>What are the 3 similar triangles? </li></ul>5.5 3.1 Y Z X W 6.3
  6. 6. Find the height of the roof <ul><li>Draw XYW and mark it. </li></ul><ul><li>Draw YZW and mark it. </li></ul><ul><li>Draw XYZ and mark it. </li></ul>5.5 3.1 Y Z X W 6.3
  7. 7. Find the height of the roof <ul><li>Use the fact that XYW ~ XZY to write a proportion: </li></ul><ul><li>YW = XY </li></ul><ul><li>ZY XZ </li></ul><ul><li>Substitute: </li></ul><ul><li>h = 3.1 </li></ul><ul><li>5.5 6.3 </li></ul><ul><li>Cross multiply: </li></ul><ul><li>6.3h = 5.5*3.1 </li></ul><ul><li>Solve: </li></ul><ul><li>h = 2.7 </li></ul>5.5 3.1 Y Z X W 6.3
  8. 8. Looking at proportions <ul><li>What is CD in ∆ADC? </li></ul><ul><li>(longer leg) </li></ul><ul><li>What is CD in ∆BCD? </li></ul><ul><li>(shorter leg) </li></ul>C D A B
  9. 9. Looking at proportions <ul><li>We can write the proportion: </li></ul><ul><li>BD = CD </li></ul><ul><li>CD AD </li></ul><ul><li>So we learn that CD is the ______ _____ of BD and AD </li></ul>C D A B
  10. 10. Geometric Means <ul><li>What other segments are in more than one triangle? </li></ul><ul><li>CB and AC </li></ul><ul><li>Their side lengths are also geometric means. </li></ul>C D A B
  11. 11. Geometric Mean Theorems <ul><li>In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into 2 segments. The length of the altitude is the geometric mean of the lengths of the 2 segments. </li></ul>C D A B
  12. 12. Geometric Mean Theorems <ul><li>In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into 2 segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. </li></ul>C D A B
  13. 13. Geometric Mean Theorems <ul><li>What proportion can you set up? </li></ul>C D A B 2 5 y
  14. 14. Geometric Mean Example <ul><li>5+2 = y </li></ul><ul><li>y 2 </li></ul><ul><li>7 = y </li></ul><ul><li>y 2 </li></ul><ul><li>y 2 = 14 </li></ul>C D A B 2 5 y
  15. 15. Example C D A B 6 3 x <ul><li>= x </li></ul><ul><li>x 3 </li></ul><ul><li>x 2 = 18 </li></ul><ul><li>X = 3√2 </li></ul>
  16. 16. How are triangles & trig important in real life?
  17. 18. Homework <ul><li>Do worksheets. </li></ul>

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