Problem:

Given a right triangle ABC with perimeter of 60 and altitude to the hypotenuse of 12, find the dimensions (lengths of the sides) of the triangle.

We know

since ABC is a right triangle. We also can conclude that

ab = 12c

since each side of the equation is twice the area of the triangle.

Consider segment DE as the side of a square and divide the square of area 3600 into nine regions as follows:

The area, 3600, is the sum of the areas of the nine regions. The area of each individual region is given in red.

But

so,

and

Continue:

Since ab = 12c

and because a + b + c = 60

72c = 1800

c = 25

Therefore

ab = 300

so

and

a = 20, b = 15

or

a = 15, b = 20

For those who see the world only through algebra, we could have begun with

then expanded and solved for a, b , and c with no reference to the areas and sub-areas.

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