What is this equation solved for \( a \) ? A \( a=\sqrt{c^{2}+b^{2}} \) B \( a=c-b \) C \( a=\frac{c^{2}-b^{2}}{2} \) D \( a=\sqrt{c^{2}-b^{2}} \) D D \( a^{2}=c^{2} \)

To determine which option correctly solves the equation for \( a \), let's first understand the context. It appears you're working with the Pythagorean theorem, which relates the sides of a right-angled triangle: \[ a^2 + b^2 = c^2 \] Here, \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides. **Objective:** Solve the equation for \( a \). **Steps to Solve for \( a \):** 1. **Start with the Pythagorean theorem:** \[ a^2 + b^2 = c^2 \] 2. **Subtract \( b^2 \) from both sides to isolate \( a^2 \):** \[ a^2 = c^2 - b^2 \] 3. **Take the square root of both sides to solve for \( a \):** \[ a = \sqrt{c^2 - b^2} \] **Matching with the Provided Options:** Let's compare this result with the given choices: - **A.** \( a = \sqrt{c^2 + b^2} \) *(Incorrect: This would apply if \( a \) were the hypotenuse.)* - **B.** \( a = c - b \) *(Incorrect: This linear equation doesn't align with the Pythagorean relationship.)* - **C.** \( a = \frac{c^2 - b^2}{2} \) *(Incorrect: The correct relationship involves a square root, not a division by 2.)* - **D.** \( a = \sqrt{c^2 - b^2} \) *(Correct: This matches our derived equation.)* **Conclusion:** **Option D** is the correct solution for \( a \): \[ a = \sqrt{c^2 - b^2} \]