8.01 Similar Right Triangles

Geometric Means Theorem

The length of the altitude drawn from the vertex of the right angle of the right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse.

A triangle with hypotenuse length h, which splits the base into lengths x and y.

Geometric Mean

This theorem allows you to find the length of a segment of the hypotenuse given the length of the altitude and the length of the other segment of the hypotenuse. You can also use this theorem to find the length of the altitude given the length of the segments of the hypotenuse.

3 step simplification. It begins with x over h equals h over y. The second is h squared equals x times y. The final is h equals the square root of ( x times y ).

Quick Quiz

Find the altutude.

A triangle with unknown altitude and base split into lengths 4 and 9.

Start by substituting in the known values.

blank over h equals blank over 9

Answer: 4, h

Use the cross product property to eliminate the fractions.

h² = (4) ( ___ )

Answer: 9

Simplify

h² = ___

Answer: 36

Take the square root of each side and simplify.

h equals the square root of 36

h = ___

Answer: 6

Similar Right Triangles

Consider the ratios of the three triangles at the beginning of the lesson.

hypotenuse over long equals (y plus x) over a equals a over x

A triangle with base split into y and x, with side a adjacent to length x

Geometric Mean Theorem

This theorem allows you to find the length of a segment of the hypotenuse given the length of the adjacent leg and the length of the other segment of the hypotenuse. You can also use this theorem to find the length of a leg of the triangle given the length of the segments of the hypotenuse.

A triangle with base split into y and x, with side a adjacent to length x

Similar Right Triangles

If the altitude is drawn to the hypotenuse of a right triangle, then the measure of a leg of the right triangle is the geometric mean between the measure of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

A triangle with base split into y and x, with side a adjacent to length x

Three step simplification. First step begins with (x plus y) over a equals a over x. Cross multiplication makes a squared equals x times (x plus y). The next step leaves a equals the square root of x times (x plus y)