The scale of a map is one
centimetre to 1.35 kilometres. The position of three towns on a
map form a triangle. Towns B and C are 17 centimetres
apart, and the angles of towns A and B are 83 degrees and 65 degrees,
respectively. Find the actual distance between
towns A and B and between towns A and C, giving the answer to the nearest
There are two things we should do
before we perform any tricky calculations. The first is to calculate the
measure of the angle at C. Since we know that angles in a
triangle add to 180 degrees, we can find the measure of the angle at C by
subtracting 83 and 65 from 180. 180 minus 83 plus 65 is 32
Next, since we’re being asked to
find the actual distances between the towns, we should convert the scale measurement
of the distance between towns B and C into the actual measurement. The scale is one centimetre to 1.35
kilometres. So we can calculate the actual
distance between towns B and C by multiplying 17 by 1.35. That gives us a distance of 22.95
Next, let’s fully label our
triangle. We know that the side opposite the
angle A can be denoted as lowercase 𝑎, the side opposite angle B is lowercase 𝑏,
and the side opposite angle C is lowercase 𝑐. So we have a non-right-angled
triangle, for which we know all three angles and the length of one of its sides.
We can use the law of sines to
calculate the lengths of the two missing sides. We know that we’re not going to use
the law of cosines since that requires at least two known side lengths. The law of sine says 𝑎 over sin A
equals 𝑏 over sin B which equals 𝑐 over sin C. Alternatively, that can be written
as sin A over 𝑎 equals sin B over 𝑏, which equals sin C over 𝑐.
We can use either of these
equations. However, since we’re trying to find
the length of the missing sides, we’ll use the first equation. By choosing this one, we’ll be
minimizing the amount of rearranging we need to do to solve our equations.
Let’s start by calculating the
distance between towns A and B. We’ve called that lowercase 𝑐. At this stage, we’re not interested
in the measure of the angle at B nor the side 𝑏. So we’re going to use these two
parts of the equation: 𝑎 over sin A equals 𝑐 over sin C.
Substituting what we know into this
formula gives us 22.95 over sin of 83 equals 𝑐 over sin of 32. Notice that we’ve used the actual
distance between the towns rather than the scale measurement. We can solve this equation by
multiplying both sides by sin of 32. 𝑐 is, therefore, equal to 22.95
over sin of 83 degrees multiplied by sin of 32, which is 12.252. Correct to the nearest kilometre,
the distance between towns A and B then is 12 kilometres.
Next, let’s calculate the distance
between towns A and C. We’ve called that lowercase 𝑏. This time, we’ll use 𝑎 over sin A
equals 𝑏 over sin B. We could have chosen to use 𝑐 over
sin C instead of 𝑎 over sin A. However, that would have involved
using some rounded answers, which we want to avoid wherever possible.
This time, when we substitute the
values into our equation, we get 22.95 over sin of 83 degrees equals 𝑏 over sin of
65 degrees. And we’re gonna solve in the exact
same way. We’re gonna multiply by sin of 65
degrees. That gives us 22.95 over sin of 83
multiplied by sin of 65, which is 20.955.
Correct to the nearest kilometre,
the distance between towns A and C is 21 kilometres.