Question

Two congruent squares are shown in figures 1 and 2 below

Answer 1

Answer:

I will answer everithing in terms of a and b.

The area of a square is equal to L*L = L^2

where L is the lenght of the side of a square.

The area of a triangle is equal to b*h/2

where b is the base of the triangle and h is the heigth.

1) Area of the triangles in the first image.

The area of one triangle is: a*b/2

then for the 4 triangles we have 4 times that amount:

Area = 4*(a*b/2) = 2*a*b

2) For the triangles in the second image we have the same situation the base is a and the heigth is b, so the area for the 4 triangles is:

Area = 4*(a*b/2) = 2*a*b

3)  The unshaded area in the first image is:

Area = c*c = c^2

And, by the Pythagorean theorem we have that:

b^2 + a^2 = c^2

Then the area of the square is:

Area = c^2 = a^2 + b^2

4) the combined areas from the two squares in image 2 are:

Area of the big one = b*b = b^2

Area of the small one = a*a = a^2

total area: a^2 + b^2

5) The total area for the two images will be:

Total area for both images = a^2 + b^2 + a^2 + b2 = 2*(a^2 + b^2)