### Video Transcript

π΄π΅πΆ is a triangle with a
perimeter of 49 centimeters where the ratio between the measure of the angle at π΄,
the measure of the angle at π΅, and the measure of the angle at πΆ is nine to five
to four. Find the length of the smallest
side, giving the answer to two decimal places.

Since the sum of the interior
angles in a triangle is 180 degrees, we can begin by sharing 180 degrees into the
ratio nine to five to four. To do this, we first find the total
number of parts by adding together each number in the ratio. Nine plus five plus four is 18. Since the measure of the angle at
π΄ is worth nine of these parts, itβs worth nine 18ths of 180 degrees. Nine 18ths simplifies to
one-half. And one-half of 180 degrees is
90. So the measure of the angle at π΄
is 90 degrees.

Similarly, the measure of the angle
at π΅ can be found by finding five 18ths of 180. We can find one 18th by dividing
180 by 18. Thatβs 10. Five 18ths is five times the size
of this. So itβs 50. And the measure of the angle at π΅
is 50 degrees. Finally, the measure of the angle
at πΆ is calculated by finding four 18ths of 180. Since one 18ths was 10 degrees,
four 18ths is four times this. Thatβs 40 degrees. So, we have a right-angled triangle
for which we know the measure of all of its angles.

Donβt be fooled into thinking we
need to use right angle trigonometry to solve this problem though. We are given that the triangle has
a perimeter of 49 centimeters. So that tells us that π plus π
plus π must be equal to 49. And, instead, weβre going to use
the law of sines to create further expressions in terms of π΄, π΅, and πΆ. 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. And this is sometimes written as
sin π΄ over π equal sin π΅ over π, which equals sin πΆ over π. We can use either of these
forms. However, since weβre trying to find
missing lengths, we should use the first form. This will reduce the amount of
rearranging we need to do.

Letβs substitute what we know into
this formula. That gives us π over sin 90 equals
π over sin 50, which equals π over sin 40. Letβs split this up a bit. Weβre trying to find the shortest
side in the triangle, which is the side opposite the smallest angle. The smallest angle is πΆ. So the side opposite that is
denoted by lowercase π. What we need to do then is form an
equation for π and π in terms of π. Letβs start with π. π over sin 90 equals π over sin
40. We know that sin 90 is equal to
one. So, in fact, π is equal to π over
sin 40. We also know that π over sin 50 is
equal to π over sin 40. We can multiply both sides of this
equation by sin 50 to get an equation for π in terms of π. Thatβs π is equal to π sin 50
over sin 40.

We now have two expressions in
terms of π, which we can substitute into our equation for the perimeter of the
triangle. Letβs clear some space. π equals π over sin 40. And π equals π sin 50 over sin
40. Which if we replace in our original
equation for the perimeter gives us π over sin 40 plus π sin 50 over sin 40 plus
π equals 49. This looks really nasty. But we can factorize by taking out
a factor of π. That gives us π multiplied by one
over sin 40 plus sin 50 over sin 40 plus one is equal to 49.

If we type the expression inside
the brackets into our calculator, we get 3.74747 and so on. We wonβt round this number just
yet. In fact, weβll use its exact form
in the next stage of our calculation. To solve this equation for π, we
can divide through by this 3.74. And doing so, we get that π is
equal to 13.075 and so on. Rounding this correct to two
decimal places and we get that the length of the shortest side, π, is 13.08
centimeters.