In the standard (x,y) coordinate plane , point A is located at `(w,w+5)` and point B is located at `(4w,w-5)` . In coordinate units , what is the distance between A and B ?

Options

A. `sqrt(9w^2+2w+10)`
B. `sqrt(9w^2+100)`
C. `9w^2+100`
D. `|w|sqrt(11)`

Answer

To find the distance between points A and B in the coordinate plane, we will use the distance formula. The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

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

### Step-by-Step Solution:

1. **Identify the coordinates of points A and B**:
- Point A is located at \( (w, w + 5) \).
- Point B is located at \( (4w, w - 5) \).

2. **Assign the coordinates**:
- Let \( x_1 = w \), \( y_1 = w + 5 \) (for point A).
- Let \( x_2 = 4w \), \( y_2 = w - 5 \) (for point B).

3. **Substitute the coordinates into the distance formula**:
\[ d = \sqrt{(4w - w)^2 + ((w - 5) - (w + 5))^2} \]

4. **Simplify the expressions**:
- Calculate \( x_2 - x_1 \):
\[ 4w - w = 3w \]
- Calculate \( y_2 - y_1 \):
\[ (w - 5) - (w + 5) = w - 5 - w - 5 = -10 \]

5. **Substitute the simplified values back into the distance formula**:
\[ d = \sqrt{(3w)^2 + (-10)^2} \]

6. **Calculate the squares**:
\[ d = \sqrt{9w^2 + 100} \]

7. **Final expression for the distance**:
\[ d = \sqrt{9w^2 + 100} \]

### Conclusion:
The distance between points A and B is \( \sqrt{9w^2 + 100} \).