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Chapter 7: Coordinate Geometry

    Introduction

    Distance Between Two Points

    Distance Between Two Points on a Line Parallel To The Coordinate Axis

    Distance Between Any Two Points In the X-Y Plane

    Section Formula

    TriSectional Points of a Line

    Centroid Triangle

    Area of the Triangle

    Straight Lines

    Summary

    Enhanced Curriculum Support

    Easy Level Worksheet

    Moderate Level Worksheet

    Hard Level Worksheet

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Chapter 7: Coordinate Geometry > Distance Between Two Points

Distance Between Two Points

Let us consider the following situation:

A town B is located 36 km east and 15 km north of the town A.

How would you find the distance from town A to town B without actually measuring it? Let us see.

This situation can be represented graphically as shown. You may use the Theorem to calculate this distance.

Consider the figure above. What is the distance from b to a?

Since both the points lie on the x-axis, we just need to walk on the x axis. So the distance is nothing but ob-oa = units.

Now, what is the distance from d to c? These points are on the y-axis and their distance is units.

Can we find the distance between a and c? The distance between a and c is units. (hint: apply pythagoras theorem).

Next, can you find the distance of A from C? Since OA = 4 units and OC = 3 units, the distance of A from C, i.e., AC = 32+42 = units.

Similarly, you can find the distance of B from D = BD = units.

Now, if we consider two points not lying on coordinate axis, can we find the distance between them? !

We shall use Pythagoras theorem to do so. Let us see an example.

This is also easy enough. We just need to visualize a scenario which we know. Draw a line from P to Q.

The line is just hanging in the air. We still don't have enough information. Why don't you draw a perpendicular from P to x-axis namely, PR?

We are getting somewhere, but still we dont have any formation that we can recognize. Lets continue. Draw a perpendicular from Q to x-axis namely, QS.

Now we are getting somewhere. We just need one more final line segment. Can you guess and draw?

!

Now you have a right angle triangle and once we have the right angle triangle we can use our ever dependable Pythogoras theorem. Using the theorem we get the distance PQ is .

Consider the points P(6, 4) and Q(–5, –3). Draw QT perpendicular to the x-axis.

Also draw a perpendicular PT from the point P on QT to meet y-axis at the point R.

A perpendicular from the point P on QS is drawn to meet it at the point T.

Then PT = units and QT = units.

Using the Pythagoras Theorem to the right triangle PTQ, we get PQ = 112+72 = units.