right triangle \( A B C \), angle \( C \) is a right angle, \( A B=17 \), and \( B C=15 \). What is the ngth of \( A C \) ? Show using the Right-Triangle Theorem formula and label \( \mathrm{a}, \mathrm{b}, \mathrm{c} \).

To find the length of side \( AC \) in right triangle \( ABC \) using the **Pythagorean Theorem** (also known as the Right-Triangle Theorem), let's follow these steps: ### 1. **Labeling the Sides** First, let's label the sides of the triangle for clarity: - **Side \( a \):** Opposite angle \( A \) → \( BC = 15 \) - **Side \( b \):** Opposite angle \( B \) → \( AC \) (the length we need to find) - **Side \( c \):** Opposite angle \( C \) (the right angle) → \( AB = 17 \) \[ \begin{align*} & \triangle ABC \\ & \text{Right Angle at } C \\ & a = BC = 15 \\ & b = AC = \text{?} \\ & c = AB = 17 \\ \end{align*} \] ### 2. **Applying the Pythagorean Theorem** The Pythagorean Theorem states that in a right-angled triangle: \[ a^2 + b^2 = c^2 \] Plugging in the known values: \[ 15^2 + b^2 = 17^2 \] \[ 225 + b^2 = 289 \] ### 3. **Solving for \( b \) (Length of \( AC \))** Subtract 225 from both sides to isolate \( b^2 \): \[ b^2 = 289 - 225 \] \[ b^2 = 64 \] Take the square root of both sides: \[ b = \sqrt{64} \] \[ b = 8 \] ### 4. **Conclusion** The length of side \( AC \) is **8 units**. \[ \boxed{8} \]