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.
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.
Quick Quiz
Find the altutude.
Start by substituting in the known values.
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 = ___
Answer: 6
Similar Right Triangles
Consider the ratios of the three triangles at the beginning of the lesson.
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.
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.