Question Video: Using the Law of Sines to Calculate an Unknown Length in a Triangle Mathematics • Second Year of Secondary School

Video Transcript

From point 𝐴 on a riverbank, a man looked at a house located on the other side of the river at point 𝐵 and found the direction to be 39 degrees north of east. He walked 147 meters east parallel to the river and arrived at point 𝐶 where point 𝐵 was 59 degrees north of east. Given the two riverbanks are parallel and points 𝐴, 𝐵, and 𝐶 are on the same horizontal level, find the width of the river giving the answer to the nearest meter.

There’s a lot of information given in the question, but helpfully we’ve been given a diagram to represent it. We can see points 𝐴 and 𝐶 on one side of the river and point 𝐵, which represents the house, on the other side. The angles of 39 degrees and 59 degrees from each of the points 𝐴 and 𝐶 to the point 𝐵 are marked on the diagram. Each of these angles were described as north of east. So considering a compass where the northerly direction is taken to be the vertical direction on our screen, north of east would be an angle measured in this direction upwards from the easterly direction. We’re also given the distance between points 𝐴 and 𝐶, which is the 147 meters the man walked in an easterly direction parallel to the river.

We’re asked to find the width of the river, which we can see marked on our diagram here. Now, we can see that it is a triangle formed by points 𝐴, 𝐵, and 𝐶. But the width of the river isn’t one of the sides of this triangle. Instead, if we were to draw in a vertical line from point 𝐵 to the extension of line 𝐴𝐶, which meets this line at a right angle, then the width would be the length of the line segment 𝐵𝐷. We can label this as 𝑤 meters, and it is a length in the right triangle 𝐵𝐶𝐷.

Currently, however, the only information we know about this triangle is that it is a right triangle with an angle of 59 degrees. We could work out the measure of the third angle in this triangle, but without the lengths of any sides, we won’t be able to go any further.

Notice though that the side 𝐵𝐶 is shared with triangle 𝐴𝐵𝐶. And in triangle 𝐴𝐵𝐶, we do know a side length. The length of 𝐴𝐶 is 147 meters. So perhaps we can do some work in triangle 𝐴𝐵𝐶 first to enable us to work out the length of side 𝐵𝐶. In triangle 𝐴𝐵𝐶 then, we know that side 𝐴𝐶 is 147 meters and angle 𝐵𝐴𝐶 is 39 degrees. We can also work out the measure of angle 𝐴𝐶𝐵 because this angle is on a straight line with the angle of 59 degrees. Angles on a straight line sum to 180 degrees. So the measure of angle 𝐴𝐶𝐵 is 180 degrees minus 59 degrees, which is 121 degrees.

We can also work out the measure of the third angle in triangle 𝐴𝐵𝐶. Angles in a triangle sum to 180 degrees. So the measure of angle 𝐴𝐵𝐶 is 180 degrees minus 39 degrees minus 121 degrees, which is 20 degrees. In triangle 𝐴𝐵𝐶, we now know the measures of all three angles and the length of one side. We want to calculate the length of a second side, side 𝐵𝐶, and we can do this using the law of sines.

This states that in any triangle 𝐴𝐵𝐶 where capital 𝐴, capital 𝐵, and capital 𝐶 represent the measures of the three angles and lowercase 𝑎, lowercase 𝑏, and lowercase 𝑐 represent the lengths of the side opposite the angle with the corresponding letter, 𝑎 over sin 𝐴 is equal to 𝑏 over sin 𝐵, which is equal to 𝑐 over sin 𝐶. The side of length 147 meters is opposite the angle of 20 degrees, and the side whose length we wish to calculate, 𝐵𝐶, is opposite the angle of 39 degrees.

So using the law of sines, we can form an equation. 𝐵𝐶 over sin of 39 degrees is equal to 147 over sin of 20 degrees. We can solve this equation to determine the length of 𝐵𝐶 by multiplying both sides by sin of 39 degrees. That gives 𝐵𝐶 equals 147 sin 39 degrees over sin of 20 degrees. We can now evaluate this on our calculator, which must be in degree mode, and it gives 270.481 continuing. Now, we’ll keep that exact value on our calculator display.

We’ve now worked out the length of the side 𝐵𝐶, which, remember, was shared with triangle 𝐵𝐶𝐷. If we return to triangle 𝐵𝐶𝐷, which is a right triangle, we now know the length of one side and the measures of all the angles. We can therefore apply right triangle trigonometry to calculate the length of the side 𝐵𝐷, which will give the width of the river.

Labelling the three sides of this triangle in relation to the angle of 59 degrees, 𝐵𝐷 is the opposite, 𝐶𝐷 is the adjacent, and 𝐵𝐶 is the hypotenuse. The side whose length we know is the hypotenuse, and the side whose length we wish to calculate is the opposite. So recalling the acronym SOH CAH TOA, it is the sine ratio that we need to use to answer this question.

For an angle 𝜃 in a right triangle, the sin of angle 𝜃 is defined to be equal to the length of the opposite side divided by the length of the hypotenuse. So substituting 59 degrees for 𝜃, 𝑤 for the opposite, and 270.481 continuing for the hypotenuse, we have sin of 59 degrees is equal to 𝑤 over 270.481 continuing. To solve for 𝑤, we multiply both sides by the denominator of 270.481 continuing. Now, if you’ve kept that exact value on your calculator, you can now just type multiplied by sin of 59 degrees to give an exact value for 𝑤. And it is 231.84 continuing.

Now, the question specifies that we should give our answer to the nearest meter. So we’re rounding to the nearest integer, and as the value in the first decimal place is an eight, we’re going to be rounding up.

So by first using the law of sines in a nonright triangle and then the sine ratio in a right triangle, we found the width of the river to the nearest meter is 232 meters.