1982 AHSME Problems/Problem 24

Solution 1

We now know that because and the whole perimeter of the triangle. And because the equilateral triangle will have side length . Harnessing the power of point , we have We also have on point : We will simplify this, by inputting values we already know.

What we know:

Side length

and

Now we know what is, We also can derive as it equals

so by the prior calculations, Therefore

We can update our chart:

What we know:

Side length 16

and

and it's pretty obvious that We can also derive Therefore However, we know the value of Because we have Therefore We can distribute: and But, we know that the side lengths must sum to so we have We can now update our 'What we Know section.

What we know:

Side length 16

and

Taking the power of point B, we have and because we already know , we know Now we shall multiply and to get Therefore

Updating our chart,

What we know:

Side length 16

and

We can now substitute and have 2 equations and two variables. and are especially useful. Because and we have or and However, because we have or and or Because the differences and are constantly and