Video: AQA GCSE Mathematics Foundation Tier Pack 3 • Paper 1 • Question 16

A treasure map shows three locations of buried treasure: 𝐴, 𝐵, and 𝐶. a) What is the three-figure bearing of location 𝐵 from location 𝐴? b) Find the actual distance between location 𝐴 and location 𝐶.

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Video Transcript

A treasure map shows three locations of buried treasure: 𝐴, 𝐵, and 𝐶. And the scale of the map is one centimeter on the map represents 60 meters in real life. Part a, what is the three-figure bearing of location 𝐵 from location 𝐴? And part b, find the actual distance between location 𝐴 and location 𝐶.

Now, when we’re finding the three-figure bearing of location 𝐵 from location 𝐴, it means that we need to stand at point 𝐴, face due north, and then rotate in a clockwise direction and measure how many degrees we need to turn before we directly face point 𝐵. And the three-figure bearing means that we have to give three digits in our answer. So, for example, if we only had to turn one degree, we’d call that zero, zero, one degrees. If we had to turn 15 degrees, it will be zero, one, five degrees. So we need to work out where is north on our diagram.

Well, the convention is that north is directly towards the top of the page. So we need to draw a north arrow, starting at point 𝐴 like this. Then, we’ll need to join point 𝐴 to point 𝐵 with a line like this. Then, we’ll need to use our protractor to measure this angle here to find out how many degrees we’d have to turn clockwise from point 𝐴 to face point 𝐵.

So on our map, we can use a nice sharp pencil and a ruler to accurately draw in these lines, the north arrow from 𝐴 and the line joining 𝐴 and 𝐵. Then, we can carefully place our protractor on the diagram so that we can measure the angle. The baseline of the protractor, the one that goes through both of the zeros should line up with the north arrow. Then, the point on the protractor which lies on the intersection between the 90-degree line and the baseline should exactly line up with point 𝐴. We’re trying to measure the size of the angle between the north line and the line between 𝐴 and 𝐵.

You’ll notice round the outside of the protractor, there are two sets of numbers. We’re interested in the set of numbers that start from the north at zero degrees not start in counting from 180 degrees. So following those numbers in a clockwise direction, we go from nought to 10 to 20 to 30, and so on, all the way round to over 110, half way between 110 and 120. In this case, that’s 115 degrees. Now, this seems to make sense. The angle that we’ve seen is somewhere between 90 degrees and 180 degrees. So this looks like it’s about the right figure.

Now, if we’d read the other numbers, the ones beginning at 180 up here, we’d have ended up with an answer of maybe 75 or 65 degrees, depending on how we misread it. But because the angle that we’ve got is over 90 degrees, we know that would’ve been wrong.

So one final thing to check. Have we got three digits? 115 — one, one, five — that does have three figures. So we do have a three-figure bearing of location 𝐵 from location 𝐴.

Now, in part b, we have to find the actual distance between locations 𝐴 and 𝐶. And now, we’re gonna take two steps. One, we’re gonna measure the distance between 𝐴 and 𝐶 on the map. And then, we’re gonna use the scale to calculate the actual real-life distance between those two points. So let’s just make a note of the scale. And then we can actually use our ruler to measure the distance between those two points. Carefully lining up the zero with point 𝐴, we can see that point 𝐶 is exactly 11 centimeters away. So now, we can do our calculation.

The scale tells us that one centimeter on the map represents 60 meters in real life. But we’ve measured a distance of 11 centimeters on the map. That’s a distance which is 11 times as long as one centimeter. That means it’s going to represent a distance which is 11 times as long as 60 meters in real life. So what’s 11 times 60? Well, 11 is 10 plus one. So 11 times 60 is 10 times 60 plus one times 60. And 10 times 60 is 600, and one times 60 is 60. So we’ve got 600 plus 60.

So the answer to part b is, the actual distance between location 𝐴 and location 𝐶 is 660 meters. And this is roughly what your map should look like by the time you’ve done all your annotation.

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