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 $$ a^{2}=c^{2} $$

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UpStudy AI · Accepted Answer

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 $$ a $$:

$$
a = \sqrt{c^2 - b^2}
$$
$$ a = \sqrt{c^{2} - b^{2}} $$, choice D

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 \( a^{2}=c^{2} \)

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