The diagram shows the path of the boat in relation to a lighthouse. The boat has sailed north from point 𝐴 and is currently at point 𝐵. When it reached point 𝑋, it was directly west of the lighthouse at point 𝐿. 𝐴𝐵 to 𝐴𝑋 equals four to three. 𝐴𝑋 equals 12.3 miles. At point 𝐵, the boat stops and turns 138 degrees clockwise to face directly towards
the lighthouse. Calculate how far the boat is from the lighthouse at point 𝐵. Give your answer to three significant figures.
So what I have actually done first with this question is mark on the diagram what
we’re actually looking to find. And that’s the distance from 𝐵 to 𝐿 — so the distance from point 𝐵 to the
lighthouse. So now, the first step is to actually use the information we’ve got to be able to
write some more values onto our diagram.
So first of all, we’ve got 𝐴𝐵 to 𝐴𝑋 is equal to four to three and 𝐴𝑋 is equal
to 12.3 miles. So therefore, what I’ve done is actually written this information down nice and
clearly. So what we can see is actually three parts are gonna be equal to 12.3 or 12.3
miles. So therefore, one part of our ratio is gonna be equal to 12.3 divided by three, which
will be equal to 4.1.
And then, what we can actually do is use this to find 𝐴𝐵 because 𝐴𝐵 is gonna be
equal to four cause it’s four parts multiplied by 4.1. So therefore, 𝐴𝐵 is gonna be equal to 16.4. So I’ve added this annotation to our diagram. So it says 16.4 miles for 𝐴𝐵. But what I can also do is actually add the distance from 𝑋 to 𝐵 because we know
from the ratio for the difference between 𝐴𝐵 and 𝐴𝑋 is one part and one part
equals 4.1, so therefore, we can say the distance from 𝑋 to 𝐵 is 4.1 miles.
So now, the next thing to do is actually look at the question again and see what
other information we’ve got. And it tells us that at point 𝐵, the boat stops and turns 138 degrees clockwise. And that’s actually to face the lighthouse. So I’ve actually also marked this on our diagram. So what we can actually say is that the bearing from 𝐵 to the lighthouse is 138
degrees. So actually, what we can do with this bit of information is add on a couple of other
So as you can see, I’ve actually added on a couple of more angles. So we’ve got 48 degrees. And we’ve got this because as you can see I’ve actually added a right angle. And that’s because if we’re going to the horizontal, it’s 90 degrees. And then if you add 48 degrees to 90 degrees, you get 138 degrees, which is the
bearing that we already mentioned.
And the way that we’ve actually found the 42 degrees is well, you could use a couple
of ways. One actually is a straight line. So angles on a straight line add up to 180 degrees. So 138 plus 42 gives 180 degrees. And also, we know that it’s half a circle. So therefore, it’s gonna be 180 degrees. And again, for the same reason you get 42 degrees.
Okay, fab! So now, what we can do is actually get on with the problem and find 𝐵𝐿. There’re actually a couple of ways to actually find out what 𝐵𝐿 is because there
are in fact two triangles we could use. So I’m gonna show you this way first and that’s using the triangle 𝐵𝑋𝐿.
I have included the information that we know which is that 𝐵𝑋 is equal to 4.1 or 4.1
miles. And we know that the angle of 𝐵 is equal to 42 degrees. And also, we also know that it’s actually a right-angled triangle. And we know that because 𝐿 is in fact due east of 𝑋. So now, take a look at our triangle. We see we’ve got a side. We also want to find one of the sides, which is 𝐵𝐿. And we have an angle.
So therefore, as it’s a right-angled triangle, we know that we’re gonna use the trig
ratios. It’s not going to be Pythagoras because obviously for Pythagoras, we’d actually need
to know at least two of the sides. Okay, great, so we know that. So now, let’s actually use that to find 𝐵𝐿.
So we are gonna use the trig ratios to find 𝐵𝐿. And the way that I like to do this is actually through these clear steps. So step one is actually label. So I want to label the sides. So first of all, we have the hypotenuse because this is actually the longest side
opposite the right angle. And then, we have the opposite. And this is because this is the opposite the angle that we’re either finding or that
we know. And then, finally, the adjacent because this is the side that’s next to the angle
that we’ve got, so adjacent to it.
Okay, now, we’ve actually labelled the sides. What we do is move on to step two and that’s which ratio. So we need to decide which ratio we’re actually going to use. So what I do now is actually circle the side we’re looking for first, which is the
hypotenuse, so 𝐵𝐿 and the side that we’ve got, which is 𝐵𝑋, which is our
So then, what I do is to take a look at SOHCAHTOA which is what we actually use to
help us remember the trig ratios. And we can see that because we have adjacent and hypotenuse, we’re gonna use the
cosine ratio because this tells us that the cosine of an angle is equal to the
adjacent divided by the hypotenuse. So now, as we completed step two, we move on to step three.
And step three is actually substitute. So what we’re gonna do is actually substitute our values in. And when we do that, what we actually get is that actually the cosine of 42 is equal
to 4.1, which is our adjacent, divided by our hypotenuse, which is 𝐵𝐿. So then, what we do is move on to our final step, which is step four which is to
rearrange and solve.
So what we do first is actually multiply each side by 𝐵𝐿. So that gives us 𝐵𝐿 cos 42 is equal to 4.1. And then if we divided each side by cos 42, we get 𝐵𝐿 is equal to 4.1 over cos
42. So therefore, we get that 𝐵𝐿 is equal to 5.517 et cetera.
Well, have we finished at this point? Well, no, there is one final stage because if we looked out at the bottom, it tells
us that it wants our answer to three significant figures. So therefore, we can say that 𝐵𝐿 is equal to 5.52 miles to three significant
figures. And we got that because if we look back at our answer we had, then one is actually
the third significant figure. And because the number after it was a seven, which is five and above, we actually
round the one to a two. So it gave us 5.52 miles to three significant figures. So this was the length of 𝐵𝐿, which was the distance of the boat from the
lighthouse when it’s at point 𝐵.
So great, we’ve actually solved the problem. But what I did say was that I was gonna show you how you could actually solve it
using another triangle. Well, the other triangle we can use actually uses the angle 48 degrees. And I have actually drawn it here. So now, what we do is actually go through the steps as before.
So the first steps is actually to label the sides, which I’ve done now. So we got the hypotenuse, the opposite, and the adjacent. Then, we actually move on to step two. And step two is actually to decide which ratio to use. Well, this time we’ve actually got the hypotenuse as what we want to find cause
that’s 𝐵𝐿. We’ve got the opposite cause we know that’s 4.1. So therefore, we’re gonna use the sine ratio or SOH, which is that the sine of an
angle is equal to the opposite divided by the hypotenuse.
So now, we move on to step three. And in step three, what we’ve actually done is substituting our values in. So we have sin 48 is equal to the opposite which is 4.1 divided by our hypotenuse
which is 𝐵𝐿. So then, what we do is step four, which is rearrange and solve. So what we do first is multiply by 𝐵𝐿 and divide by sin 48. So we get that 𝐵𝐿 is equal to 4.1 over sin 48. So again, we actually get the answer 5.52 to three significant figures.
So therefore, great, we’ve checked the answer and we’ve actually shown that you can
find it using two different methods. So we know that the distance from the boat to the lighthouse when the boat is at
point 𝐵 is 5.52 miles to three significant figures.