1. **State the problem:** We are given a rectangle ABCD with sides DC and AD defined as functions of $x$: $$DC = \frac{5}{x^2 - 25}$$ $$AD = \frac{3}{x - 5}$$ We want to understand these expressions and possibly simplify or analyze them. 2. **Recall the formula for the area of a rectangle:** $$\text{Area} = \text{length} \times \text{width}$$ Here, the sides DC and AD represent the length and width. 3. **Simplify the expressions:** Note that $x^2 - 25$ is a difference of squares: $$x^2 - 25 = (x - 5)(x + 5)$$ So, $$DC = \frac{5}{(x - 5)(x + 5)}$$ 4. **Express the area:** $$\text{Area} = DC \times AD = \frac{5}{(x - 5)(x + 5)} \times \frac{3}{x - 5} = \frac{15}{(x - 5)^2 (x + 5)}$$ 5. **Important domain restrictions:** - $x \neq 5$ because it makes denominators zero. - $x \neq -5$ for the same reason. 6. **Summary:** - Side DC is $\frac{5}{(x - 5)(x + 5)}$ - Side AD is $\frac{3}{x - 5}$ - Area is $\frac{15}{(x - 5)^2 (x + 5)}$ These expressions describe the rectangle's sides and area as functions of $x$ with domain restrictions to avoid division by zero.