Question

the diagram shows three squares that are joined at vertices to form a right triangle. which statement is true? a the sum of the areas of square n and square k is equal to the area of square l. b the sum of the areas of square n and square l is greater than the area of square k. c the sum of the areas of square n and square k is less than the area of square l. d the sum of the areas of square n and square l is equal to the area of square k.

Explanation:

Step1: Recall Pythagorean theorem

If the side - lengths of the right - triangle are \(a\), \(b\), and \(c\) (where \(c\) is the hypotenuse), then \(a^{2}+b^{2}=c^{2}\). The area of a square with side - length \(s\) is \(A = s^{2}\). Let the side - length of Square \(N\) be \(a\), the side - length of Square \(L\) be \(b\), and the side - length of Square \(K\) be \(c\). Since the three squares are joined to form a right - triangle, by the Pythagorean theorem, \(a^{2}+b^{2}=c^{2}\). And the area of Square \(N=a^{2}\), the area of Square \(L = b^{2}\), and the area of Square \(K=c^{2}\).

Step2: Determine the relationship between the areas

The sum of the areas of Square \(N\) and Square \(L\) is \(a^{2}+b^{2}\), and the area of Square \(K\) is \(c^{2}\). So, the sum of the areas of Square \(N\) and Square \(L\) is equal to the area of Square \(K\).

Answer:

D. The sum of the areas of Square N and Square L is equal to the area of Square K.