Hi, there--

The Problem:
The area of a right triangle is 50. One of its angles is 45 degrees. What are the lengths of 
the sides and hypotenuse of the triangle?

A Solution:
The sum of the interior angles of any triangle is 180 degrees. We have a right triangle, so 
one angle has a measure of 90 degrees. We are told that another angle measures 45
degrees. The remaining angle also has a measure of 45 degrees because 90+45+45=180.

The sides opposite two equal angles are equal in measure, so we have a right triangle with legs 
of equal length. These form the base and height of the triangle.

Let's find the length of the legs of the triangle. The formula for the area of a triangle is "one half the base times the height," or B*H/2.

We are given that the area is 50, so
B*H/2 = 50

Multiply both sides of this equation by 2.
B*H = 50*2 = 100

Recall that the base and height are equal in this triangle, so substitute B for H in the equation.
B*B = 100

B = 10 because 10 times itself equals 100.

The base is 10. The height is also 10 because the base and height are equal. Therefore, the
legs of this triangle are 10 and 10.

Now let's find the length o the hypotenuse.
This is a right triangle so we can apply the Pythagorean Equation, , where
a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse.



Substitute 10 for a and 10 for b in the equation.



Simplify.



Take the square root of both sides.


The length of the hypotenuse is exactly . This is an irrational number. If you use 
your calculator to express your answer as a decimal, you will only have an approximation. 
Leave your answer as  to be exact.

Good luck!
Mrs. Figgy