Question

In Figure 1, the area of square A is 9 square units, the area of square B is 16 square units, and the area of square C is 25 square units. In Figure 2, the area of square D is 36 square units and the area of square E is 64 square units, and the area of square F is 100 square units. In Figure 3, the area of square G is 25 square units and the area of square H is 144 square units, and the area of square I is 169 square units. What can be concluded about the areas of the squares? A. The sum of the areas of the two smaller squares is less than the area of the larger square. B. The sum of the areas of the two smaller squares is greater than the area of the larger square. C. The sum of the areas of the two smaller squares is equal to the area of the larger square. D. The sum of the areas of the two smaller squares is equal to the area of the triangle.

Answer 1

A. The sum of the areas of the two smaller squares is equal to the area of the larger square.

Explanation:

In Figure 1, a, b, and c form the sides of PQR. They also form the sides of square A, square B, and square C respectively.

For a right triangle, the Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Applying the Pythagorean theorem to PQR, it can be seen that a2 + b2 = c2.

Similarly, the following statements will be true for XYZ and STU.

For XYZ, d2 + e2 = f2.

For STU, g2 + h2 = i2.

By observation, along with the Pythagorean theorem, the sum of the areas of the two smaller squares is equal to the area of the larger square.

Answer 2

C. The sum of the areas of the two smaller squares is equal to the area of the larger square.

Explanation:

9 + 16 = 25

36 + 64 = 100

25 + 144 = 169

The relations "less than" and "greater than" can be ruled out. These observations are consistent with selection C.

The triangle area is half the product of the square roots of the squares on the legs, so the areas of the triangles are (respectively) 6, 24, 30. These are not related to the sum of the smaller squares, so the last selection can also be ruled out.