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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. + 2. 5. 3.

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**DRILL Find the geometric mean between the two given numbers.**

and 8 and 4

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DRILL Find the geometric mean between the two given numbers. 6 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**

LEGS A & The perpendicular side HYPOTENUSE B C The side opposite the right angle

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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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**a:h = h:b State the means and the extremes in the following statement. **

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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**applying the law of proportion.**

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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**∆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

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**TRY THIS OUT! ∆ACD ~ ∆ABC ∆ACD ~ ∆CBD ∆ABC ~ ∆CBD D B C A**

NAME ALL SIMILAR TRIANGLES ∆ACD ~ ∆ABC ∆ACD ~ ∆CBD ∆ABC ~ ∆CBD

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**In a right triangle, the altitude to the hypotenuse is the geometric **

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 In the figure, D**

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

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

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Section 8-1 Similarity in Right Triangles. Geometric Mean If a, b, and x are positive numbers and Then x is the geometric mean. x and x are the means.

Section 8-1 Similarity in Right Triangles. Geometric Mean If a, b, and x are positive numbers and Then x is the geometric mean. x and x are the means.

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