If the distance between the points `(4, gamma)` and (1, 0) is 5, then `gamma` equals.

Options

A. 4 only
B. `-4` only
C. `+-4`
D. 0

Answer

To solve the problem, we need to find the value of `gamma` given that the distance between the points (4, gamma) and (1, 0) is 5. We will use the distance formula to do this.

### Step-by-Step Solution:

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

2. **Identify the Points**:
Here, we have:
- Point A: \( (4, \gamma) \) where \( x_1 = 4 \) and \( y_1 = \gamma \)
- Point B: \( (1, 0) \) where \( x_2 = 1 \) and \( y_2 = 0 \)

3. **Set Up the Equation**:
According to the problem, the distance \( d \) is 5. Therefore, we can write:
\[ 5 = \sqrt{(1 - 4)^2 + (0 - \gamma)^2} \]

4. **Square Both Sides**:
To eliminate the square root, we square both sides of the equation:
\[ 5^2 = (1 - 4)^2 + (0 - \gamma)^2 \]
This simplifies to:
\[ 25 = (-3)^2 + (-\gamma)^2 \]
Which further simplifies to:
\[ 25 = 9 + \gamma^2 \]

5. **Rearrange the Equation**:
Now, we will isolate \( \gamma^2 \):
\[ \gamma^2 = 25 - 9 \]
This gives us:
\[ \gamma^2 = 16 \]

6. **Solve for \( \gamma \)**:
Taking the square root of both sides, we find:
\[ \gamma = \pm 4 \]

### Final Answer:
Thus, the value of \( \gamma \) is \( 4 \) or \( -4 \).