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 the given points, we will 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.

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

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

2. **Substitute the coordinates into the distance formula**:
\[ d_1 = \sqrt{(2 - (-6))^2 + (-2 - 4)^2} \]

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

4. **Square the differences**:
\[ d_1 = \sqrt{(8)^2 + (-6)^2} \]
\[ d_1 = \sqrt{64 + 36} \]

5. **Add the squares**:
\[ d_1 = \sqrt{100} \]

6. **Take the square root**:
\[ d_1 = 10 \]

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

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

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

2. **Substitute the coordinates into the distance formula**:
\[ d_2 = \sqrt{(0 - (-5))^2 + (4 - (-1))^2} \]

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

4. **Square the differences**:
\[ d_2 = \sqrt{(5)^2 + (5)^2} \]
\[ d_2 = \sqrt{25 + 25} \]

5. **Add the squares**:
\[ d_2 = \sqrt{50} \]

6. **Simplify the square root**:
\[ d_2 = \sqrt{25 \times 2} = 5\sqrt{2} \]

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