Question

1. one leg of an isosceles right triangle has a length of less than 8 inches. which of the following could not be the length of the hypotenuse of the triangle? a) 2 b) 5 c) 9 d) 12

Explanation:

Step1: Recall the formula for the hypotenuse of an isosceles right triangle

In an isosceles right triangle, if the length of each leg is \( a \), then by the Pythagorean theorem, the hypotenuse \( c \) is given by \( c = \sqrt{a^2 + a^2}=\sqrt{2a^2}=a\sqrt{2} \).

Step2: Determine the maximum possible length of the hypotenuse

Given that one leg \( a < 8 \) inches. Substitute \( a = 8 \) (the upper bound, not included) into the formula for the hypotenuse. So \( c = 8\sqrt{2}\approx8\times1.414 = 11.312 \) inches.

Step3: Analyze each option

• Option A: \( 2 \). If \( c = 2 \), then \( a=\frac{2}{\sqrt{2}}=\sqrt{2}\approx1.414<8 \), so this is possible.
• Option B: \( 5 \). If \( c = 5 \), then \( a=\frac{5}{\sqrt{2}}\approx3.535<8 \), so this is possible.
• Option C: \( 9 \). If \( c = 9 \), then \( a=\frac{9}{\sqrt{2}}\approx6.364<8 \), so this is possible.
• Option D: \( 12 \). Since the maximum possible hypotenuse (when \( a \) approaches 8) is approximately \( 11.312 \), and \( 12>11.312 \), so \( 12 \) could not be the length of the hypotenuse.

Answer:

D) 12