Question

one leg of a right - triangle is 7 inches shorter than the other leg. when is the hypotenuse at least 17 inches long? (1 point) when the longer leg is greater than - 8 inches and less than 15 inches. when the longer leg is greater than 0 inches and less than 15 inches. when the longer leg is greater than or equal to 15 inches. when the longer leg is less than or equal to - 8 inches or greater than or equal to 15 inches.

Explanation:

Step1: Let the shorter leg be $x$ inches, then the longer leg is $x + 7$ inches. Let the hypotenuse be $c$ inches. By the Pythagorean theorem, $c^{2}=x^{2}+(x + 7)^{2}=x^{2}+x^{2}+14x + 49=2x^{2}+14x + 49$. We want to find when $c\geq17$, so $c^{2}\geq289$. Then $2x^{2}+14x + 49\geq289$.

Step2: Rearrange the inequality to get a quadratic - inequality: $2x^{2}+14x-240\geq0$. Divide through by 2: $x^{2}+7x - 120\geq0$.

Step3: Factor the quadratic equation $x^{2}+7x - 120=(x + 15)(x - 8)=0$. The roots are $x=-15$ and $x = 8$.

Step4: Consider the quadratic function $y=x^{2}+7x - 120$. The parabola opens upward (since the coefficient of $x^{2}$ is positive). The inequality $x^{2}+7x - 120\geq0$ is satisfied when $x\leq - 15$ or $x\geq8$. But since $x$ represents the length of a side of a triangle, $x>0$. So we focus on $x\geq8$. The longer leg is $x + 7$, so when $x\geq8$, the longer leg $x + 7\geq15$.

Answer:

When the longer leg is greater than or equal to 15 inches.