Km is a straight line of 13 units. If K has the co-ordinates (2,5) and M has the co-ordinates (x, -7), find the values of x.

Answer

To solve the problem, we need to find the value of \( x \) such that the distance between the points \( K(2, 5) \) and \( M(x, -7) \) is 13 units. We will use the distance formula, which is given by:

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

### Step-by-Step Solution:

1. **Identify the coordinates**:
- Point \( K \) has coordinates \( (2, 5) \) (where \( x_1 = 2 \) and \( y_1 = 5 \)).
- Point \( M \) has coordinates \( (x, -7) \) (where \( x_2 = x \) and \( y_2 = -7 \)).

2. **Set up the distance equation**:
- The distance \( d \) between points \( K \) and \( M \) is given as 13 units.
- Therefore, we can write:
\[ \sqrt{(x - 2)^2 + (-7 - 5)^2} = 13 \]

3. **Simplify the equation**:
- Calculate \( -7 - 5 \):
\[ -7 - 5 = -12 \]
- Substitute this back into the equation:
\[ \sqrt{(x - 2)^2 + (-12)^2} = 13 \]
- This simplifies to:
\[ \sqrt{(x - 2)^2 + 144} = 13 \]

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

5. **Isolate the squared term**:
- Subtract 144 from both sides:
\[ (x - 2)^2 = 169 - 144 \]
- This simplifies to:
\[ (x - 2)^2 = 25 \]

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

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

8. **Conclusion**:
- The values of \( x \) are \( 7 \) and \( -3 \).
- Therefore, the coordinates of point \( M \) can be \( (7, -7) \) or \( (-3, -7) \).

### Final Answer:
The values of \( x \) are \( 7 \) and \( -3 \).

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