Question

Select the correct answer from each drop-down menu. This square has the hypotenuse of a right triangle as one of its sides. The estimated value of the length of the hypotenuse is ______ units. The estimated value of the area of the square is ______ square units. (Hint: The hypotenuse is the side opposite the right angle. To find the hypotenuse of a right triangle, use the Pythagorean theorem, c² = a² + b².)

1) Cannot be determined
2) 2 units
3) 4 units
4) 8 units
5) 16 units
6) 32 units
7) 64 units

Answer 1

The estimated value of the length of the hypotenuse of the right triangle is approximately 10.3 units, and therefore, the estimated value of the area of the square is approximately 106.09 square units.

Step-by-step explanation:

The student's question involves finding the estimated length of the hypotenuse of a right triangle and then calculating the area of the square whose side is this hypotenuse. Using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): c2 = a2 + b2. Based on the provided example, if the lengths of sides a and b are 9 units and 5 units respectively, the length of the hypotenuse is calculated as c = √(92 + 52) = √(81 + 25) = √106 ≈ 10.3 units.

Therefore, the estimated value of the length of the hypotenuse is approximately 10.3 units. Since the area of the square is given by A = c2, and c is approximately 10.3, the estimated value of the area is A = (10.3)2 ≈ 106.09 square units. To match given multiple-choice options and considering significant figures, the length of the hypotenuse can be estimated as 10 units and the area of the square as 100 square units.