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.

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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.

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Quick Quiz

Find the altutude.

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Start by substituting in the known values.

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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.

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h = ___

Answer: 6

Similar Right Triangles

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

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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.

See more: What Is Next In This Series? 1, 4, 10, 19, 31, _ ? Centered Triangular Number

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.