Video: AQA GCSE Mathematics Higher Tier Pack 2 โ€ข Paper 2 โ€ข Question 7

Use a ruler and a compass to answer this question. ๐ด, ๐ต, and ๐ถ are three points. Another point, ๐‘ƒ, is 4 cm away from ๐ด. It is also the same distance from ๐ต and ๐ถ. Mark one of the possible positions of point ๐‘ƒ.

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Video Transcript

Use a ruler and a compass to answer this question. ๐ด, ๐ต, and ๐ถ are three points. Another point, ๐‘ƒ, is four centimetres away from ๐ด. It is also the same distance from ๐ต and ๐ถ. Mark one of the possible positions of point ๐‘ƒ.

Now itโ€™s important to say that, for this question, you need to print it at 100-percent scale. Donโ€™t use the scale-to-fit paper option when printing out the question. If you print it out at a slightly different size, because this is a scale drawing question involving constructions, your answer will look a little bit different to ours.

Weโ€™re told to use a ruler and compass to answer this question. And that makes it an accurate construction question. That means we got to make marks on the page and not rub them out. Thatโ€™s our working out. If we rub out all of our working out lines, then we may lose marks.

Our diagram then gives us three points: ๐ด, ๐ต, and ๐ถ. And we have to work out where another point, ๐‘ƒ, is, which is four centimetres away from ๐ด. And weโ€™re also told that point ๐‘ƒ is the same distance away from points ๐ต and ๐ถ. Sometimes thatโ€™s written as being equidistant from ๐ต and ๐ถ.

Our first task then is to get the compasses and ruler out. Put the compass point exactly on zero mark on the ruler. Then carefully open up the radius until itโ€™s exactly four centimetres. Then with the radius set to exactly four centimetres, we can place the compass point exactly on point ๐ด and draw a circle, centre ๐ด, with radius four centimetres. Then check carefully with your ruler that the radius of this circle is indeed exactly four centimetres.

Then to find the set of points which are equidistant from points ๐ต and ๐ถ, we need to put the compass point at ๐ถ, then open up the radius to over half the distance from ๐ถ to ๐ต. Now it doesnโ€™t matter exactly what that radius is so long as itโ€™s over half the distance from ๐ถ to ๐ต.

Then we carefully draw an arc. Whatever radius you used, itโ€™s important that you take great care to keep that radius exactly the same for the next step. And with that same radius, carefully put the compass point at ๐ต and draw another arc intersecting the first arc that you drew.

Now provided you made sure that the radius of your compasses was over half the distance between ๐ต and ๐ถ, then you will be able to create a second arc so that it cuts the first arc in two places. If we wouldโ€™ve gone for a slightly smaller radius, our arcs wouldโ€™ve intersected slightly close together. And if weโ€™d gone for a slightly larger radius, theyโ€™d intersect a little bit further apart.

Now draw a line between your points of intersection and extend it through the circle around point ๐ด. Now when youโ€™ve done this, youโ€™ll find that all of the points on this line are equidistant from points ๐ต and ๐ถ. In other words, any point on that line is the same distance away from point ๐ต as it is from point ๐ถ.

And remember, we said that all of the points on this circle are four centimetres away from ๐ด. If you look carefully, you can see that there are two points which are both on that circle of points which are four centimetres away from ๐ด and also on the line of points which are equidistant from ๐ต and ๐ถ. So we got two possible positions for our point ๐‘ƒ.

The question asked us to mark one of the possible positions of point ๐‘ƒ. So either one of those is a correct answer. And finally, if your printer did print this off at a different scale, then your position of ๐‘ƒ may be slightly different to our position of ๐‘ƒ. But hopefully, going through this video has helped you to work out what the steps are in answering this question.

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