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SIMILARITIES IN A RIGHT TRIANGLE

SIMILARITIES IN A RIGHT TRIANGLE. By: SAMUEL M. GIER. How much do you know. DRILL. SIMPLIFY THE FOLLOWING EXPRESSION. 1. 4. + 2. 5 . 3. DRILL.

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SIMILARITIES IN A RIGHT TRIANGLE

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  1. SIMILARITIES IN A RIGHT TRIANGLE By: SAMUEL M. GIER

  2. How much do you know

  3. DRILL • SIMPLIFY THE FOLLOWING EXPRESSION. 1. 4. + 2. 5. 3.

  4. DRILL • Find the geometric mean between the two given numbers. 1. 6 and 8 2. 9 and 4

  5. DRILL • Find the geometric mean between the two given numbers. • 6 and 8 h= = = h= 4

  6. DRILL • Find the geometric mean between the two given numbers. 2. 9 and 4 h= = h= 6

  7. REVIEW ABOUT RIGHT TRIANGLES A LEGS & The perpendicular side HYPOTENUSE B C The side opposite the right angle

  8. SIMILARITIES IN A RIGHT TRIANGLE By: SAMUEL M. GIER

  9. CONSIDER THIS… State the means and the extremes in the following statement. 3:7 = 6:14 The means are 7 and 6 and the extremes are 3 and 14.

  10. CONSIDER THIS… State the means and the extremes in the following statement. 5:3 = 6:10 The means are 3 and 6 and the extremes are 5 and 10.

  11. CONSIDER THIS… State the means and the extremes in the following statement. a:h = h:b The means are h and the extremes are a and b.

  12. CONSIDER THIS… Find h. a:h = h:b applying the law of proportion. h² = ab h= his the geometric mean between a & b.

  13. THEOREM:SIMILARITIES IN A RIGHT TRIANGLE States that “In a right triangle, the altitude to the hypotenuse separates the triangle into two triangles each similar to the given triangle and to each other.

  14. M S R O ∆MOR ~ ∆MSO, ∆MOR ~ ∆OSR by AA Similarity postulate) ILLUSTRATION • “In a right triangle (∆MOR), the altitude to the hypotenuse(OS) separates the triangle into two triangles(∆MOS & ∆SOR )each similar to the given triangle (∆MOR)and to each other. ∆MSO~ ∆OSR by transitivity

  15. A B D C TRY THIS OUT! • NAME ALL SIMILAR TRIANGLES ∆ACD ~ ∆ABC ∆ACD ~ ∆CBD ∆ABC ~ ∆CBD

  16. COROLLARY 1. In a right triangle, the altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse

  17. A B D C ILLUSTRATION • CB is the geometric mean between AB & BD. In the figure,

  18. COROLLARY 2. In a right triangle, either leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to it.

  19. A B D C ILLUSTRATION • CB is the geometric mean between AB & BD. In the figure,

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