Find the distance between the points (a,b) and `(-a, -b)`.

Answer

To find the distance between the points \( (a, b) \) and \( (-a, -b) \), we can 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 the first point be \( P_1 = (a, b) \), so \( x_1 = a \) and \( y_1 = b \).
- Let the second point be \( P_2 = (-a, -b) \), so \( x_2 = -a \) and \( y_2 = -b \).

2. **Substitute the Coordinates into the Distance Formula**:
\[ d = \sqrt{((-a) - a)^2 + ((-b) - b)^2} \]

3. **Simplify the Expressions**:
- Calculate \( x_2 - x_1 \):
\[ -a - a = -2a \]
- Calculate \( y_2 - y_1 \):
\[ -b - b = -2b \]

4. **Substitute Back into the Distance Formula**:
\[ d = \sqrt{(-2a)^2 + (-2b)^2} \]

5. **Calculate the Squares**:
- \( (-2a)^2 = 4a^2 \)
- \( (-2b)^2 = 4b^2 \)

6. **Combine the Results**:
\[ d = \sqrt{4a^2 + 4b^2} \]

7. **Factor Out the Common Term**:
\[ d = \sqrt{4(a^2 + b^2)} \]

8. **Simplify the Square Root**:
\[ d = 2\sqrt{a^2 + b^2} \]

### Final Answer:
The distance between the points \( (a, b) \) and \( (-a, -b) \) is:
\[ d = 2\sqrt{a^2 + b^2} \]