If the distance between two points `(0,-5)` and `(x,0)` is 13. Find x

Answer

To solve the problem, we need to find the value of \( x \) given that the distance between the points \( (0, -5) \) and \( (x, 0) \) is 13.

### Step-by-Step Solution:

1. **Identify the Points**:
Let point A be \( (0, -5) \) and point B be \( (x, 0) \).

2. **Use the Distance Formula**:
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \( (x_1, y_1) = (0, -5) \) and \( (x_2, y_2) = (x, 0) \).

3. **Substitute the Points into the Formula**:
Substitute the coordinates into the distance formula:
\[ 13 = \sqrt{(x - 0)^2 + (0 - (-5))^2} \]
This simplifies to:
\[ 13 = \sqrt{x^2 + 5^2} \]

4. **Simplify the Equation**:
Since \( 5^2 = 25 \), we have:
\[ 13 = \sqrt{x^2 + 25} \]

5. **Square Both Sides to Eliminate the Square Root**:
Squaring both sides gives:
\[ 13^2 = x^2 + 25 \]
This simplifies to:
\[ 169 = x^2 + 25 \]

6. **Isolate \( x^2 \)**:
Subtract 25 from both sides:
\[ x^2 = 169 - 25 \]
Simplifying this gives:
\[ x^2 = 144 \]

7. **Take the Square Root**:
Taking the square root of both sides, we find:
\[ x = \pm \sqrt{144} \]
Therefore:
\[ x = \pm 12 \]

### Final Answer:
The values of \( x \) are \( 12 \) and \( -12 \).

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