Problem Solving

In right triangle ABC, AC is the hypotenuse. If AcC is 40 and AB + BC = 50, what is the area of the triangle ABC?

A. 225
B. 450
C. 25√2
D. 200
E. 200√2

The OA is A.

I'm confused with this PS question. Please, can any expert assist me with it? Thanks in advanced.

LUANDATO wrote:In right triangle ABC, AC is the hypotenuse. If AcC is 40 and AB + BC = 50, what is the area of the triangle ABC?

A. 225
B. 450
C. 25√2
D. 200
E. 200√2

The OA is A.

I'm confused with this PS question. Please, can any expert assist me with it? Thanks in advanced.

We have a right triangle whose hypotenuse is 40, and whose sides sum to 50. We want the area.

Let's call the two sides of the triangle x and y. The area will be xy/2.

We know x + y = 50, and if 40 is the hypotenuse
x^2 + y^2 = 40^2.

Let's take x + y = 50 and square both sides to get (x +y)^2 = 50^2. We can expand to get x^2 + 2xy + y^2 = 50^2.

We want xy/2. Look at our two equations
1) x^2 + 2xy + y^2 = 50^2.
2) x^2 + y^2 = 40^2

Notice that if we subtract the second from the first, we'll be left with an xy term. Terms in red will cancel out.

x^2 + 2xy + y^2 = 50^2.
- x^2 + y^2 = 40^2
____________________________
2xy = 50^2 - 40^2 ---> difference of squares!

2xy = (50 + 40)(50 - 40)
2xy = 90 * 10 = 900
xy = 450
xy/2 = 225.

The answer is A