Find the distance between each of the following pairs of points. P(-5, 7), Q(-1, 3)

Explanation:- 

To find the distance between two points in a 2D plane, we use the distance formula, which is given by:

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

where P(x1, y1) and Q(x2, y2) are the coordinates of the two points.

Using this formula, we can find the distance between the points P(-5, 7) and Q(-1, 3) as follows:

First, we substitute the coordinates of the two points into the distance formula to obtain:

[PQ = \sqrt{(-1 – (-5))^2 + (3 – 7)^2}]

Next, we simplify the expression inside the square root by performing the subtraction and squaring operations:

[PQ = \sqrt{(4)^2 + (-4)^2}]

We simplify further by performing the addition and square root operations:

[PQ = \sqrt{16 + 16}]

[PQ = \sqrt{32}]

Finally, we simplify by factoring out the largest perfect square, which is 16, from 32:

[PQ = \sqrt{16 \times 2}]

[PQ = \sqrt{16} \times \sqrt{2}]

[PQ = 4\sqrt{2}]

Therefore, the distance between the points P(-5, 7) and Q(-1, 3) is 4√2 units.