Question

Use the image to answer the question.

A figure shows a square made from four right-angled triangles that all have the same dimensions. Each of the four right angled triangles have a height labeled a, a base labeled b, and a hypotenuse labeled c. Sides a and b are positioned so that the right angle creates the four outer corners of the outer square. Each vertex of the inner square divides each side of the outer square in two unequal parts labeled a and b, where a is the shorter part and b is the longer part. Each side of the inner square, labeled c, becomes the hypotenuse of the four right-angled triangles formed at the corners of the outer square. The four right-angled triangles are shaded.

Explain a proof of the Pythagorean Theorem using the image.

(4 points)

Answer 1

Place four congruent right triangles with legs a and b and hypotenuse c inside a larger square so their right angles occupy the four corners of the large square; their hypotenuses form a smaller, tilted inner square of side c. The side of the large square equals a + b, so its area is
(a + b)^2.

The large square is partitioned into the four triangles plus the inner square, so its area also equals
4·(1/2·a·b) + c^2 = 2ab + c^2.

Equate the two expressions and simplify:
(a + b)^2 = 2ab + c^2
a^2 + 2ab + b^2 = 2ab + c^2
a^2 + b^2 = c^2.

Thus the sum of the squares of the legs equals the square of the hypotenuse (Pythagoras).