Question

incline mats, or triangle mats, are offered with different levels of incline to help gymnasts learn basic moves. as the name may suggest, two sides of the mat are right - triangles. if the height of the mat is 7 inches shorter than the length of the mat and the hypotenuse is 1 inch longer than the length of the mat, what is the length of the mat? answer 2 points length of the mat = inches

Explanation:

Step1: Define variables

Let the length of the mat be $x$ inches. Then the height of the mat is $(x - 7)$ inches and the hypotenuse is $(x+1)$ inches.

Step2: Apply Pythagorean theorem

According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a=x - 7$, $b = x$ and $c=x + 1$. So we have $(x - 7)^{2}+x^{2}=(x + 1)^{2}$.

Step3: Expand the equations

Expand each term: $(x^{2}-14x + 49)+x^{2}=x^{2}+2x + 1$.

Step4: Simplify the equation

Combine like - terms: $x^{2}-14x + 49+x^{2}-x^{2}-2x - 1=0$, which simplifies to $x^{2}-16x + 48=0$.

Step5: Factor the quadratic equation

Factor $x^{2}-16x + 48$ as $(x - 4)(x - 12)=0$.

Step6: Solve for x

Set each factor equal to zero: $x-4=0$ gives $x = 4$, and $x - 12=0$ gives $x=12$. But if $x = 4$, then the height $x-7=4 - 7=-3$ which is not possible for a length. So we reject $x = 4$.

Answer:

$12$