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 21 inches shorter than the length of the mat and the hypotenuse is 3 inches longer than the length of the mat, what is the length of the mat?

answer
length of the mat = inches

Explanation:

Define variables for the right triangle

Using the Pythagorean Theorem Applications knowledge point
Let \(x\) be the length of the mat in inches.
The height of the mat is \(x - 21\).
The hypotenuse is \(x + 3\).

Set up the equation

Using the Pythagorean Theorem Applications knowledge point

$$ (x - 21)^2 + x^2 = (x + 3)^2 $$

Expand and simplify the equation

Using the Solving Quadratic Equations knowledge point

$$
LATEXBLOCK0
$$

Solve the quadratic equation

Using the Solving Quadratic Equations knowledge point

$$
LATEXBLOCK1
$$

Determine the valid physical solution

Using the Pythagorean Theorem Applications knowledge point
Since the height \(x - 21\) must be positive:

$$
LATEXBLOCK2
$$

Answer:

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 21 inches shorter than the length of the mat and the hypotenuse is 3 inches longer than the length of the mat, what is the length of the mat?

Length of the mat = <blank>36</blank> inches