SIMILARITIES IN A RIGHT TRIANGLE

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SIMILARITIES IN A RIGHT TRIANGLE
By: SAMUEL M. GIER

How much do you know

DRILL SIMPLIFY THE FOLLOWING EXPRESSION. + 2. 5. 3.

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

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

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

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

SIMILARITIES IN A RIGHT TRIANGLE
By: SAMUEL M. GIER

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.

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.

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.

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.

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.

∆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

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

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

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

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

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