Question

question the length of one leg of a right triangle is 3 feet more than the other leg. if the hypotenuse is 15 feet, find the length of each leg. provide your answer below: feet, feet feedback more instruction submit content attribution

Explanation:

Step1: Let one leg be $x$ feet

Then the other leg is $x + 3$ feet. Apply Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a=x$, $b=x + 3$ and $c = 15$. So we have $x^{2}+(x + 3)^{2}=15^{2}$.

Step2: Expand the equation

Expand $(x + 3)^{2}=x^{2}+6x + 9$. The equation becomes $x^{2}+x^{2}+6x + 9=225$. Combine like - terms: $2x^{2}+6x+9 - 225=0$, which simplifies to $2x^{2}+6x - 216=0$. Divide through by 2: $x^{2}+3x - 108=0$.

Step3: Solve the quadratic equation

For the quadratic equation $x^{2}+3x - 108=0$, where $a = 1$, $b = 3$ and $c=-108$. Use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. First, calculate the discriminant $\Delta=b^{2}-4ac=(3)^{2}-4\times1\times(-108)=9 + 432=441$. Then $x=\frac{-3\pm\sqrt{441}}{2}=\frac{-3\pm21}{2}$.
We get two solutions for $x$: $x=\frac{-3 + 21}{2}=\frac{18}{2}=9$ and $x=\frac{-3 - 21}{2}=\frac{-24}{2}=-12$. Since length cannot be negative, we discard $x=-12$.

Step4: Find the lengths of the legs

If $x = 9$, then one leg is $x=9$ feet and the other leg is $x + 3=9+3 = 12$ feet.

Answer:

9, 12