### Video Transcript

πΏππ is a triangle where measure the measure of the angle at πΏ is equal to 54
degrees and 30 minutes, the measure of the angle at π is equal to 23 degrees and 30
minutes, and ππΏ is equal to 16.4 centimetres. Find the lengths of ππ and πΏπ, giving the answer to one decimal place.

Letβs begin by sketching a diagram of this triangle. Remember a sketch doesnβt need to be to scale, but it should be roughly in proportion
so we can check the suitability of any answers we get. Here, we have a non-right-angled triangle, for which we know the measure of two of
its angles and the length of one of its sides. We can rule out the law of cosines then since that requires a minimum of two known
sides.

Instead, weβll use the law of sines to calculate the missing lengths in this
triangle. The law of sines says that π over sin π΄ is equal to π over sin π΅ which is equal
to π over sin πΆ. Alternatively, thatβs often written as sin π΄ over π equals sin π΅ over π which
equals sin πΆ over π.

Remember itβs absolutely fine to use either of these formulae. However, since weβre trying to find missing lengths, weβll use the first one. This will prevent the amount of rearranging we need to do to solve the equations. The second formula is more suited for finding missing angles.

Next, weβll label the sides of our triangle. The side opposite the angle πΏ is given by lowercase π, the side opposite the angle
π is given by lowercase π, and the side opposite the angle π is lowercase π. At this stage, since we know the length of the side that weβve called lowercase π,
weβll need to also work out the measure of the angle at π.

Remember angles in a triangle add to 180 degrees. So we can calculate the measure of the angle at π by subtracting the known angles
from 180 degrees. Now, many calculators have a button that will allow us to write degrees in terms of
minutes and seconds. If this is the case, then we can pop this straight into our calculator as
written. However, if itβs not the case, itβs also useful to know how to change from degrees,
minutes, and seconds into decimal form.

Remember minutes and seconds are base 60. So we can write 54 degrees and 30 minutes as 54 and thirty sixtieths of a degree. Thirty sixtieths is the same as 0.5. So 54 degrees and 30 minutes is the same as 54.5 degrees. Similarly, 23 degrees and 30 minutes is equivalent to 23 and a half degrees. We can replace these in our equation to find the measure of the angle at π. That gives us 180 minus 23.5 plus 54.5, which is equal to 102 degrees.

The very final thing weβll do before calculating the lengths of ππ and πΏπ is to
change the letters in our formula for the law of sins to suit our triangle. It becomes π over sin πΏ equals π over sin π which equals π over sin π. Remember we usually only need to use two parts of this at any given time.

If we begin by finding the length of the side ππ, which we called lowercase π,
then we can use these two parts: π over sin πΏ equals π over sin π. We use these as opposed to π over sin π because we already know the length of the
side labelled with lowercase π. Substituting the known values into our formula gives us π over sin of 54.5 equals
16.4 over sin of 102.

Remember it is absolutely fine to change 54.5 degrees into 54 degrees and 30 seconds
as long as we remember to use the appropriate buttons on our calculator. To solve this equation, weβll multiply both sides by sin of 54.5. That gives us that π is equal to 16.4 over sin of 102 multiplied by sin of 54.5. And typing that into our calculator gives us that π is equal to 13.649 and so
on. Correct to one decimal place, the length ππ is 13.6 centimetres.

Next, to find the length of πΏπ, which we called lowercase π, weβll use these two
parts of the law of sines: π over sin π equals π over sin π. Substituting what we know into this formula gives us 16.4 over sin of 102 equals π
over sin of 23.5.

And we can solve this equation by multiplying both sides by sin of 23.5. That gives us π is equal to 16.4 over sin of 102 multiplied by sin of 23.5 which is
6.685. Correct to one decimal place, that gives us that the length of πΏπ is 6.7
centimetres.