Question

\

$$\begin{aligned} &x + y = 28 \\\\ &y = 28 - x \\\\ &x^2 + y^2 = h^2 \\\\ &x^2 + (28 - x)^2 = 20^2 \\\\ &x^2 + (28 - x)^2 = 400 \\\\ &x^2 + (784 - 56x + x^2) = 400 \\\\ &2x^2 - 56x + 384 = 0 \\\\ &x^2 - 28x + 192 = 0 \\\\ &ac = 192, \\quad b = -28 \\\\ &(x - 16)(x - 12) = 0 \\end{aligned}$$

Explanation:

Set up the system of equations

Using the Systems of Equations and Pythagorean Theorem Applications knowledge points

$$
LATEXBLOCK0
$$

Expand and simplify the quadratic equation

Using the Systems of Equations knowledge point

$$
LATEXBLOCK1
$$

Factor the quadratic trinomial

Using the Factoring Quadratics knowledge point

$$
LATEXBLOCK2
$$

Solve for the side lengths

Using the Systems of Equations knowledge point

$$
LATEXBLOCK3
$$

Answer:

The solutions for the side lengths of the right triangle are:

$$x = 16\text{ cm},\ y = 12\text{ cm}$$

or

$$x = 12\text{ cm},\ y = 16\text{ cm}$$