Find the distance between the points (2, 3) and (4, 1).

Answer

To find the distance between the points (2, 3) and (4, 1), 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:**
- Let \( (x_1, y_1) = (2, 3) \)
- Let \( (x_2, y_2) = (4, 1) \)

2. **Substitute the coordinates into the distance formula:**
\[ d = \sqrt{(4 - 2)^2 + (1 - 3)^2} \]

3. **Calculate \( x_2 - x_1 \) and \( y_2 - y_1 \):**
- \( x_2 - x_1 = 4 - 2 = 2 \)
- \( y_2 - y_1 = 1 - 3 = -2 \)

4. **Square the differences:**
\[ (x_2 - x_1)^2 = 2^2 = 4 \]
\[ (y_2 - y_1)^2 = (-2)^2 = 4 \]

5. **Add the squared differences:**
\[ d = \sqrt{4 + 4} = \sqrt{8} \]

6. **Simplify \( \sqrt{8} \):**
\[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \]

7. **Final answer:**
The distance between the points (2, 3) and (4, 1) is \( 2\sqrt{2} \).