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Understanding Geometric Mean in Right Triangles

Learn about geometric mean in right triangles, including properties, examples, and applications. Identify similar triangles, calculate means and extremes. Understand the theorem and corollaries related to right triangles.

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Understanding Geometric Mean in Right Triangles

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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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