Question

Find x. Simplify completely.
A
B
X
64
x = [?]
36

Answer 1

x = 48

Explanation:

You want to know the altitude of a right triangle when that segment divides the hypotenuse into parts that are 64 and 36 units.

Geometric mean

The length of x in this configuration is the geometric mean of the two lengths of the hypotenuse it creates:

x = √(64·36) = (√64)(√36) = 8·6

x = 48

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Additional comment

This comes about as a result of the similarity of all of the triangles. The ratios of long side to short side are the same, so ...

AD/DB = BD/DC

64/x = x/36

x² = 64·36

x = √(64·36) = 48

There are two other geometric mean relations, resulting from the ratios of hypotenuse to long side, and hypotenuse to short side.

You may notice these are all 3-4-5 right triangles.

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Answer 2

when running a line, in a right-triangle, from the 90° angle perpendicular to its opposite side, we will end up with three similar triangles, one Small, one Medium and a containing Large one.

Check the picture below.

`\cfrac{x}{36}=\cfrac{64}{x}\implies x^2=(36)(64)\implies x^2=(6^2)(8^2)\implies x^2=(6\cdot 8)^2 \\\\\\ x^2=48^2\implies x=√(48^2)\implies x=48`