Question

the diagram shows squares d, e, and f joined at vertices to form a right triangle. which statement is true? a the sum of the areas of square d and square f is equal to the area of square e. b the sum of the areas of square d and square e is greater than the area of square f. c the sum of the areas of square d and square e is equal to the area of square f. d the sum of the areas of square d and square f is less than the area of square e.

Explanation:

Step1: Recall Pythagorean theorem

If the side - lengths of the squares D, E, and F are \(a\), \(b\), and \(c\) respectively, and the right - triangle has legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), then by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). The area of square D is \(A_D = a^{2}\), the area of square E is \(A_E=b^{2}\), and the area of square F is \(A_F = c^{2}\).

Step2: Determine the relationship between the areas

Since \(a^{2}+b^{2}=c^{2}\), we have \(A_D + A_E=A_F\). That is, the sum of the areas of Square D and Square E is equal to the area of Square F.

Answer:

C. The sum of the areas of Square D and Square E is equal to the area of Square F.