𝐺 and 𝐻 are two points on a map. One centimetre represents 20 metres. There is a path which runs equidistant to both points 𝐺 and 𝐻. Hugo is standing at point 𝐻 on the map. He walks directly towards point 𝐺 until he reaches the path. He then turns 90 degrees and walks a distance of 120 metres and stops at a new point,
𝑋. Mark the two possible positions of point 𝑋 on the map.
In this question, we need to work accurately. And we need to work using the scale that we’ve been given. This question is a constructions question. So before we begin, you need to make sure you have all the right equipment. You need a pencil, a ruler, and a compass. Marks will be given for accuracy as well as for the correct method. So take your time and make sure you perform each of the steps we’re going to work
through carefully. Let’s have a look at the information in the question step by step.
We’re told first of all that there is a path which is equidistant from both points 𝐺
and 𝐻, which just means it’s the same distance away from these two points. Let’s think about how to draw this path in. We need to find the set or locus of all the points which are the same distance from
points 𝐺 and 𝐻.
First, we draw in the straight line connecting points 𝐺 and 𝐻 together. To construct the locus or set of all points that are at the same distance from the
end points of a line, we construct what’s known as the perpendicular bisector of
this line. Perpendicular means that it meets the line joining points 𝐺 and 𝐻 at right
angles. And bisector means it cuts this line exactly in half.
We don’t just draw this line by eye or by measuring the midpoint and using a
protractor to make a right angle. Instead, we need to use our compass to accurately construct this line. We start by placing the pointed end of our compass at one end of the line. So we’ll start at point 𝐺. We then move the arms of our compass so that they are slightly wider than half the
length of this line.
It doesn’t matter exactly how wide we set our compass to be, but it must be over half
the length of this orange line that we’ve drawn. Don’t make it too big; otherwise, your construction may be too big for the page. Now, I’ve measured this line and I make it to be 6.5 centimetres. So I’m gonna set my compass to be about three and a half centimetres.
Now, keeping our compass point at 𝐺, we sweep the pencil part around and draw an
arc. We don’t need to draw a full circle, just an arc like the one I’ve got here in
pink. Now, this is really important: don’t change the width of your compass, but carefully
move it so that the point of the compass is now at point 𝐻.
We then sweep out another arc with the pencil like the second one I’ve got here. And we should find that the two arcs cross at two points. The perpendicular bisector of that orange line which connected 𝐺 and 𝐻 is found by
drawing a straight line through the two points where the arcs cross. So that’s the line I’ve drawn in blue.
Remember you must not change the width of your compass between drawing these two
arcs. Now, this straight line is the locus of all the points which are the same distance
from points 𝐺 and 𝐻. So this line is the path.
Now, don’t on any circumstances rub out the arcs that we used in this construction,
even if you think it will make you work look neater. These arcs are proof of the method that we’ve used to create this perpendicular
bisector and they’re essential to being awarded the marks for this question. You may perhaps want to press slightly lighter with your pencil when you draw
them. But they must still be visible at the end of the question.
Right, we found our path. The next part of the question tells us that Hugo is standing at point 𝐻 on the
map. We’re then told that he walks directly towards point 𝐺 until he reaches the
path. Now, if he is walking directly towards point 𝐺, this means that he is going to be
taking the shortest route, which is the route that is perpendicular or to right
angle to the path. This means that Hugo is walking along the orange line. He stops when he reaches the path. So now, he’s standing at the point that I’ve marked as 𝑌.
Finally, we’re told that Hugo turns to a 90 degrees and then walks a distance of 120
metres until he stops at a new point 𝑋. We are asked to find the two possible positions of 𝑋. The reason there are two possible positions is because we don’t know whether Hugo
turns clockwise or anticlockwise. So we need to draw on both possibilities.
Now, remember we need to do this accurately. We’re told in the question that one centimetre represents 20 metres. In order to mark the position of 𝑋 accurately, we need to know how many centimetres
on the map will represent 120 metres. Well, 120 divided by 20 is six, which means to scale up from 20 metres to 120 metres,
we need to multiply by six.
Multiplying one centimetre by six gives six centimetres. So this tells us that six centimetres on our map represents 120 metres in real
life. We need to measure six centimetres in either direction along the path from point
𝑌. Take your time and be accurate about this. There isn’t much room for error in questions like this. So you really need to make sure you are measuring six centimetres and not just
something close to six centimetres.
I’ve marked in one of the positions for 𝑋. And now, measuring in the other direction, I’ve marked in the second position for
𝑋. The two possible positions of point 𝑋 are now marked on the map.
Remember the arcs that we drew in constructing our perpendicular bisector are an
essential part of our answer to this question.