Video: Using the Sine Rule to Calculate Unknown Lengths in a Triangle

The scale of a map is 1 cm : 1.35 km. The position of three towns on a map form a triangle. Towns B and C are 17 cm apart, and the angles of towns A and B are 83ยฐ and 65ยฐ respectively. Find the actual distance between towns A and B and between towns A and C giving the answer to the nearest kilometre.

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Video Transcript

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 kilometre.

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 degrees.

Next, since weโ€™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 kilometres.

Next, letโ€™s fully label our triangle. We know that the side opposite the angle A can be denoted as lowercase ๐‘Ž, the side opposite angle B is lowercase ๐‘, and the side opposite angle C is lowercase ๐‘. 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 weโ€™re not going to use the law of cosines since that requires at least two known side lengths. The law of sine says ๐‘Ž over sin A equals ๐‘ over sin B which equals ๐‘ over sin C. Alternatively, that can be written as sin A over ๐‘Ž equals sin B over ๐‘, which equals sin C over ๐‘.

We can use either of these equations. However, since weโ€™re trying to find the length of the missing sides, weโ€™ll use the first equation. By choosing this one, weโ€™ll be minimizing the amount of rearranging we need to do to solve our equations.

Letโ€™s start by calculating the distance between towns A and B. Weโ€™ve called that lowercase ๐‘. At this stage, weโ€™re not interested in the measure of the angle at B nor the side ๐‘. So weโ€™re going to use these two parts of the equation: ๐‘Ž over sin A equals ๐‘ over sin C.

Substituting what we know into this formula gives us 22.95 over sin of 83 equals ๐‘ over sin of 32. Notice that weโ€™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. ๐‘ 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, letโ€™s calculate the distance between towns A and C. Weโ€™ve called that lowercase ๐‘. This time, weโ€™ll use ๐‘Ž over sin A equals ๐‘ over sin B. We could have chosen to use ๐‘ over sin C instead of ๐‘Ž 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 ๐‘ over sin of 65 degrees. And weโ€™re gonna solve in the exact same way. Weโ€™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.

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