A right triangle has sides A, B, and C. Side A is the hypotenuse and side B is also a side of a rectangle. Sides A, C, and the side of the rectangle adjacent to side B have lengths of #13 #, #9 #, and #3 #, respectively. What is the rectangle's area?

1 Answer
Dean R.
Apr 25, 2018

# 6 sqrt{22} #

Explanation:

As is my practice, I'm converting to standard notation where triangle sides are small letters #a,b,c#. While we're naming things, let's call the other side of the rectangle #d#.

A right triangle has hypotenuse #a# and sides #b#, #c#. Side #b# forms a rectangle with another length #d#. Given #a=13, c=9, d=3# what is the area of the rectangle?

The area we seek is #bd# so we have to find #b#, so this is all a long windup to a Pythagorean Theorem question:

#a^2 = b^2 + c^2#

#a# is the hypotenuse so it's the one all by itself.

#b^2= a^2 - c^2 = 13^2-9^2 = 169-81 = 88#

#b = sqrt{88} = 2 sqrt{22}#

# text{area} = bd = 6 sqrt{22} #

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