### Video Transcript

Cities π΄, π΅, and πΆ are located such that city π΄ is west of city π΅, city πΆ is on a bearing of 35 degrees from city π΅, and city πΆ is 100 miles from city π΄ and 70 miles from city π΅. Find the distance between cities π΄ and π΅ giving your answer to one decimal place.

Letβs begin by sketching a diagram. Remember this doesnβt need to be to scale. However, it should be roughly in proportion so that we can check the suitability of any answers we get. Weβll start by adding city π΅ and a north line to our diagram. City π΄ is some distance west of city π΅. That means that the angle between the north line and the line π΄π΅ must be 90 degrees.

City πΆ is on a bearing of 35 degrees from city π΅. Remember bearings are measured in a clockwise direction from the north line. So the angle between the north line and π΅πΆ is 35 degrees. We know that city πΆ is 100 miles from city π΄ and 70 miles from city π΅. We can calculate the measure of the angle at π΅ by adding together 90 and 35. 90 plus 35 is 125 degrees. So the measure of the angle at π΅ is 125.

Weβre trying to calculate the distance between city π΄ and π΅. Weβll call that side lowercase π because it sits directly opposite the angle πΆ. In order to use the law of sines to calculate the length of the side π, weβd first need to know the measure of the angle at πΆ.

What we can do first though is calculate the measure of the angle at π΄ since we know the length of the side π. Letβs use the law of sines then to calculate the measure of this angle. We can use either of these versions. But since weβre first trying to find the missing length, itβs sensible to use the first one. This will minimise the amount of rearranging we need to do.

Since weβre trying to calculate the measure of the angle at π΄ and we know the measure of the angle at π΅, weβll use the first two parts of the formula: sin π΄ over π equals sin π΅ over π. Substituting what we know into this formula gives us sin π over 70 equals sin 125 over 100.

We can solve this equation by multiplying both sides by 70 to get sine of π equals 70 sine of 125 all over 100. That gives us a value of 0.573. To calculate the value of π then, weβll find the inverse sine of both sides of this equation. π is equal to the inverse sine of 0.573 which is 34.988 degrees. We wonβt round this answer just yet; instead, weβll use its exact form in our future calculations.

Since we know that angles in a triangle add to 180 degrees, we can calculate the measure of the angle at πΆ by subtracting the two known angles from 180. The measure of the angle at πΆ then is 20.011 degrees. And now that we know the measure of the angle at πΆ, we can use the sine rule once again to find the length labelled π.

Remember since weβre finding a side, we can now use the second version for the law of sines. Weβll use π over sin π΅ equals π over sin πΆ. Substituting what we know into this formula gives us 100 over sine of 125 equals π over sine of 20.011.

Weβll solve this equation by multiplying both sides by sine of 20.011. π is equal to 100 over sine of 125 multiplied by sine of 20.011 which is 41.776.

Correct to one decimal place, the distance between cities π΄ and π΅ is 41.8 miles.