Maths

Line segments - Higher

A line segment is part of a line which has two end points (ie, it is not infinite). Each end of a line segment is usually labelled with letters.

The length of a line segment

If A is the point (1, 1) and B is the point (4, 5), what is the length of the line segment AB?

It is not easy to picture this without drawing a sketch. Have a look at the diagram below:

image: diagonal line on a line graph

A has an x-coordinate of 1. B has an x-coordinate of 4. So, to get from A to B, we move along 3 units.

A has a y-coordinate of 1. B has a y-coordinate of 5. So, to get from A to B, we move up 4 units.

We have created a right-angled triangle. So to find the length of AB we use Pythagoras’ theorem.

AB² = 3² + 4²

AB² = 25

AB = square root of 25

AB = 5

Question

If P is the point (1, 5) and Q is the point (5, 1), what is the length of the line segment PQ? Give your answer correct to 1 decimal point (d.p.).

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Answer
image: diagonal line on graph

PQ² = 4² + 4²

PQ² = 32

PQ = square root of 32

AB = 5.7 (1 d.p.)

Note: If you get a question like this, it is fine to draw a sketch or diagram. If you get really confident, you might be able to answer the question without using a diagram.

The midpoint of a line segment

If X is the point (1, 1) and Y is the point (3, 5), what is the midpoint of the line segment XY?

Look at the diagram below:

image: diagonal line on graph

It is clear from this diagram that the midpoint of (1, 1) and (3, 5) is (2, 3).

In fact, the x-coordinate of M is the average of the x-coordinates of X and Y

And the y-coordinate of M is the average of the y-coordinates of X and Y.

So:M = (1 + 3 over 2, 1 + 5 over 2) = (2, 3)

Question

What is the midpoint of (2, 7) and (4, 1)?

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Answer

The midpoint of (2, 7) and (4, 1) is:

(2 + 4 over 2, 7 + 1 over 2) = (3, 4)

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