Find the value of p if the distance between the points (4, p) and (1, 0) is 5.

Options

A. `pm 4`
B. `pm 6`
C. `pm 8`
D. `pm 7`

Answer

To find the value of \( p \) such that the distance between the points \( (4, p) \) and \( (1, 0) \) is 5, we can use the distance formula.

### Step-by-step solution:

1. **Identify the points**:
Let point A be \( (4, p) \) and point B be \( (1, 0) \).

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

3. **Substitute the coordinates**:
Here, \( x_1 = 4 \), \( y_1 = p \), \( x_2 = 1 \), and \( y_2 = 0 \). Thus, the distance can be expressed as:
\[ d = \sqrt{(1 - 4)^2 + (0 - p)^2} \]

4. **Set the distance equal to 5**:
According to the problem, the distance is 5. Therefore, we have:
\[ \sqrt{(1 - 4)^2 + (0 - p)^2} = 5 \]

5. **Square both sides to eliminate the square root**:
\[ (1 - 4)^2 + (0 - p)^2 = 5^2 \]
Simplifying this gives:
\[ (-3)^2 + (-p)^2 = 25 \]
\[ 9 + p^2 = 25 \]

6. **Isolate \( p^2 \)**:
Subtract 9 from both sides:
\[ p^2 = 25 - 9 \]
\[ p^2 = 16 \]

7. **Solve for \( p \)**:
Taking the square root of both sides gives:
\[ p = \pm 4 \]

Thus, the possible values of \( p \) are \( 4 \) and \( -4 \).

### Final Answer:
The values of \( p \) are \( 4 \) and \( -4 \).

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