If A = (x,-7 ) , B = (2,5) and AB = 13 units , findx.

Answer

To solve the problem, we need to find the value of \( x \) given the points \( A = (x, -7) \), \( B = (2, 5) \), and the distance \( AB = 13 \) units. We will use the distance formula, which is given by:

\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step-by-Step Solution:

1. **Identify the coordinates:**
- \( A = (x, -7) \) where \( x_1 = x \) and \( y_1 = -7 \)
- \( B = (2, 5) \) where \( x_2 = 2 \) and \( y_2 = 5 \)

2. **Substitute into the distance formula:**
\[ AB = \sqrt{(2 - x)^2 + (5 - (-7))^2} \]
Since \( AB = 13 \), we can set up the equation:
\[ \sqrt{(2 - x)^2 + (5 + 7)^2} = 13 \]

3. **Simplify the equation:**
\[ \sqrt{(2 - x)^2 + 12^2} = 13 \]
\[ \sqrt{(2 - x)^2 + 144} = 13 \]

4. **Square both sides to eliminate the square root:**
\[ (2 - x)^2 + 144 = 169 \]

5. **Isolate the squared term:**
\[ (2 - x)^2 = 169 - 144 \]
\[ (2 - x)^2 = 25 \]

6. **Take the square root of both sides:**
\[ 2 - x = 5 \quad \text{or} \quad 2 - x = -5 \]

7. **Solve for \( x \):**
- For \( 2 - x = 5 \):
\[ -x = 5 - 2 \]
\[ -x = 3 \implies x = -3 \]
- For \( 2 - x = -5 \):
\[ -x = -5 - 2 \]
\[ -x = -7 \implies x = 7 \]

8. **Final values of \( x \):**
The possible values for \( x \) are:
\[ x = -3 \quad \text{or} \quad x = 7 \]