Question

I'm stuck on this homework question. I forgot the formula to find the side. This is 9th grade geometry.

Answer 1

x = 2root(22)

see image.

Explanation:

The triangle shown is one big triangle cut into two more smaller triangles: one medium-sized and one smaller.

ALL THREE TRIANGLES ARE SIMILAR BY AA.

Set the two smaller triangles up so you can see the corresponding sides. x is the short leg in one triangle and it is the long leg in the smallest triangle. Set up a proportion.

22/x = x/4

crossmultiply

x^2 = 22•4

x^2 = 88

square root both sides.

x = sqroot(88)

x = 2sqroot(22)

see image.

Answer 2

x = 2√22

Explanation:

You want the altitude x that divides a right triangle's hypotenuse into segments of lengths 22 and 4.

Similar triangles

All of the right triangles in the figure are similar, so they all have the same ratio of long leg to short leg:

22/x = x/4

88 = x² . . . . . . multiply by 4x

x = √88

x = 2√22 . . . . . simplified

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Additional comment

This shows that the segment marked x is the "geometric mean" of the segments of the hypotenuse: x = √(22·4). There are other "geometric mean" relations involving the legs of the largest triangle, and the segments of the hypotenuse:

long leg = √(22·(22+4)) = 2√143

short leg = √(4·(22+4)) = 2√26

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