The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. By applying the Pythagorean Theorem to square ABCD and using the fact that square MNOP is formed by joining the midpoints of square ABCD, we can prove the relationship between the side lengths of the two squares.The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.In the given image, we can see that square ABCD has side lengths AM, MB, BN, NC, CO, OD, DP, and PA equal to a and b.The square MNOP is formed by joining the midpoints of square ABCD, therefore its sides are 'c' each.Using this information, we can apply the Pythagorean Theorem to square ABCD.The diagonal of square ABCD is equal to the hypotenuse of a right triangle with sides a and b.Therefore, according to the Pythagorean Theorem,Substituting the length of the diagonal as 2c, we haveIn simplified form, the Pythagorean Theorem in this case becomes, which proves the relationship between the side lengths of square ABCD and the square MNOP....

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