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SIMILARITIES IN A RIGHT TRIANGLE By: SAMUEL M. GIER

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How much do you know

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DRILL SIMPLIFY THE FOLLOWING EXPRESSION

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DRILL Find the geometric mean between the two given numbers and and 4

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DRILL Find the geometric mean between the two given numbers and 8 h= = = h= 4

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DRILL Find the geometric mean between the two given numbers and 4 h= = h= 6

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REVIEW ABOUT RIGHT TRIANGLES A CB LEGS & HYPOTENUSE The side opposite the right angle The perpendicular side

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SIMILARITIES IN A RIGHT TRIANGLE By: SAMUEL M. GIER

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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.

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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.

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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.

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

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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.

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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. M S R O MOR ~ MSO, MOR ~ OSR by AA Similarity postulate) MSO ~ OSR by transitivity

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

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

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

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

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

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