Question

first up, lets review the assignments learning objectives get familiar with this topic by reviewing instruction and answering a couple of questions. question a right triangles hypotenuse is 10 inches long. the length of one leg of a right triangle is 2 inches more than the second leg. what are the lengths of the legs of the triangle? provide your answer below: inches, inches

Explanation:

Step1: Let the length of one leg

Let the length of the shorter leg be $x$ inches. Then the length of the other leg is $x + 2$ inches.

Step2: Apply Pythagorean theorem

According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse and $a,b$ are the legs. So $x^{2}+(x + 2)^{2}=10^{2}$.

Step3: Expand the equation

Expand $(x + 2)^{2}=x^{2}+4x + 4$. The equation becomes $x^{2}+x^{2}+4x + 4=100$.

Step4: Combine like - terms

Combine the $x^{2}$ terms: $2x^{2}+4x+4 = 100$. Then subtract 100 from both sides to get a quadratic equation: $2x^{2}+4x-96 = 0$. Divide through by 2: $x^{2}+2x - 48=0$.

Step5: Solve the quadratic equation

Factor the quadratic equation: $(x + 8)(x - 6)=0$. Set each factor equal to zero: $x+8 = 0$ gives $x=-8$ (rejected since length cannot be negative), $x - 6=0$ gives $x = 6$.

Step6: Find the lengths of the legs

If $x = 6$, then the other leg is $x + 2=6 + 2=8$.

Answer:

6 inches, 8 inches