Find the distance between the following points : (i) `A(-6, 4) and B(2, -2)` (ii) `A(-5, -1) and B(0,4)`

Answer

To find the distance between two points in a coordinate plane, we use the distance formula:

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

where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

### Step-by-Step Solution:

#### Part (i): Find the distance between points A(-6, 4) and B(2, -2)

1. **Identify the coordinates**:
- Point A: \( (x_1, y_1) = (-6, 4) \)
- Point B: \( (x_2, y_2) = (2, -2) \)

2. **Substitute the coordinates into the distance formula**:
\[ d = \sqrt{(2 - (-6))^2 + (-2 - 4)^2} \]

3. **Calculate the differences**:
- \( x_2 - x_1 = 2 - (-6) = 2 + 6 = 8 \)
- \( y_2 - y_1 = -2 - 4 = -6 \)

4. **Square the differences**:
\[ d = \sqrt{(8)^2 + (-6)^2} \]
\[ d = \sqrt{64 + 36} \]

5. **Add the squares**:
\[ d = \sqrt{100} \]

6. **Take the square root**:
\[ d = 10 \]

Thus, the distance between points A(-6, 4) and B(2, -2) is **10 units**.

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#### Part (ii): Find the distance between points A(-5, -1) and B(0, 4)

1. **Identify the coordinates**:
- Point A: \( (x_1, y_1) = (-5, -1) \)
- Point B: \( (x_2, y_2) = (0, 4) \)

2. **Substitute the coordinates into the distance formula**:
\[ d = \sqrt{(0 - (-5))^2 + (4 - (-1))^2} \]

3. **Calculate the differences**:
- \( x_2 - x_1 = 0 - (-5) = 0 + 5 = 5 \)
- \( y_2 - y_1 = 4 - (-1) = 4 + 1 = 5 \)

4. **Square the differences**:
\[ d = \sqrt{(5)^2 + (5)^2} \]
\[ d = \sqrt{25 + 25} \]

5. **Add the squares**:
\[ d = \sqrt{50} \]

6. **Simplify the square root**:
\[ d = \sqrt{25 \times 2} = 5\sqrt{2} \]

Thus, the distance between points A(-5, -1) and B(0, 4) is **\( 5\sqrt{2} \) units**.

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