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 28 inches shorter than the length of the mat and the hypotenuse is 4 inches longer than the length of the mat, what is the length of the mat? answer length of the mat = inches

Explanation:

Step1: Let the length of the mat be $x$ inches.

Then the height of the mat is $(x - 28)$ inches and the hypotenuse is $(x + 4)$ inches.

Step2: Apply the Pythagorean theorem.

Since it's a right - triangle, $a^{2}+b^{2}=c^{2}$, where $a=x - 28$, $b=x$, and $c=x + 4$. So, $(x - 28)^{2}+x^{2}=(x + 4)^{2}$.

Step3: Expand the equations.

$(x^{2}-56x + 784)+x^{2}=x^{2}+8x + 16$.

Step4: Simplify the equation.

$x^{2}-56x + 784+x^{2}-x^{2}-8x - 16 = 0$, which simplifies to $x^{2}-64x+768 = 0$.

Step5: Solve the quadratic equation.

For a quadratic equation $ax^{2}+bx + c = 0$ ($a = 1$, $b=-64$, $c = 768$), we can use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ or factor. Factoring gives $(x - 16)(x - 48)=0$.

Step6: Find the solutions.

Setting each factor equal to zero: $x - 16=0$ gives $x = 16$, and $x - 48=0$ gives $x = 48$. But if $x = 16$, then the height $x-28=16 - 28=-12$, which is not possible for a length. So we reject $x = 16$.

Answer:

48