Find the distances between the following points. (i) A(a, 0), B(0, a)

Explanation:-

The given problem provides us with two points in a 2-dimensional space, namely point A and point B. The coordinates of point A are (a, 0), which means that it is located on the x-axis, a units away from the origin (0,0). Similarly, the coordinates of point B are (0, a), which means that it is located on the y-axis, a units away from the origin.

The problem asks us to find the distance between points A and B, denoted as AB. To do this, we need to use the distance formula in 2-dimensional space, which tells us that the distance between two points (x1, y1) and (x2, y2) is given by:

[d = \sqrt{(x2-x1)^2 + (y2-y1)^2}]

Using this formula, we can find the distance between points A and B as follows:

[AB = \sqrt{(0-a)^2 + (a-0)^2}]

Here, we have substituted the coordinates of A and B into the distance formula. Simplifying the expression inside the square root, we get:

[AB = \sqrt{a^2 + a^2}]

Using the fact that a^2 + a^2 = 2a^2, we can further simplify this expression:

[AB = \sqrt{2a^2}]

Finally, we can factor out the square root of 2 from under the radical to get the final answer:

[AB = a\sqrt{2}]

This tells us that the distance between points A and B is a times the square root of 2.