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First, we need to find the area of the trapezoid by using the area formula of the trapezoid.
A=(1/2)h(b1+b2) area of a trapezoid
In the above diagram, h=a+b, b1=a, and b2=b.
A=(1/2)(a+b)(a+b)
=(1/2)(a^2+2ab+b^2).
Now let's find the area of the trapezoid by summing the area of the three right triangles.
The area of the yellow triangle is
A=1/2(ba).
The area of the red triangle is
A=1/2(c^2).
The area of the blue triangle is
A= 1/2(ab).
The sum of the area of the triangles is
1/2(ba) + 1/2(c^2) + 1/2(ab) = 1/2(ba + c^2 + ab) = 1/2(2ab + c^2).
Since, this area is equal to the area of the trapezoid we have the following relation:
(1/2)(a^2 + 2ab + b^2) = (1/2)(2ab + c^2).
Multiplying both sides by 2 and subtracting 2ab from both sides we get a^2+b^2=c^2, concluding the proof.
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