The **midpoint** of a **line segment** is the halfway point between the two ends of a line. You can work it out by adding the two `x`-coordinates together and dividing by two to get the `x`-coordinate of the midpoint; and then repeat with the `y`-coordinate.

The formula for the midpoint of a line segment is `(frac(x_1+x_2)(2), frac(y_1 + y_2)(2))`.

What is the midpoint between (3,11) and (7,15)?

Add the two `x`-coordinates then divide by two = `frac(3 + 7)(2)` = 5

Add the two `y`-coordinates then divide by two = `frac(11 + 15)(2)` = 13

Midpoint coordinate is (5, 13)

Answer: (5, 13)

A line segment is drawn from (-5,-10) to (1,-4). A perpendicular line passes through the midpoint of this line segment.

What is the value of the `y`-coordinate on the perpendicular line when `x` = 0?

Work out the equation of the line segment as `y=mx+x`

gradient = `frac(6)(6)` = 1

For the point (-5, -10)

`-10 = 1 xx -5 + c`

`c = -5`

Check with other point: -4 = 1 x 1 - 5 ✔

Line segment has equation `y=x - 5`

Midpoint of line segment: `x=frac(-5 + 1)(2) = -2`

And `y = frac(-10 + -4)(2) = -7`

Midpoint is therefore (-2, -7)

The perpendicular line has a gradient of `frac(-1)(m) = frac(-1)(1) = -1`

Perpendicular line passes through the point (-2, -7) and substituting

`-7 = -1 x -2 + c` gives `c=-9`

And the equation of the perpendicular line is `y = -x - 9`

When `x` = 0, `y` = 0 - 9

Answer: -9

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