# Question Video: Finding the Length of a Side of a Triangle given Its Area- the Length of a Side- and the Measure of an Angle Mathematics

π΄π΅πΆ is a triangle where π΄π΅ = 18 cm, πβ π΅ = 60Β°, and the area of the triangle is 74β3 cmΒ³. Find length π΅πΆ giving the answer to two decimal places.

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### Video Transcript

π΄π΅πΆ is a triangle where π΄π΅ equals 18 centimeters, the measure of angle π΅ equals 60 degrees, and the area of the triangle is 74 root three square centimeters. Find length π΅πΆ, giving the answer to two decimal places.

Letβs begin by sketching this triangle as best we can. We know that we have a side of length 18 centimeters, so hereβs that side. And then the measure of the angle at π΅ is 60 degrees. We can draw in a line that meets the side π΄π΅ at an angle of 60 degrees, but we donβt know how long this side should be. There are lots of possibilities at this stage then for what the triangle looks like. It could look like either of these two possibilities. Or, in fact, the side π΅πΆ could be extended off the screen, and the angle at π΄ could be an obtuse angle. The other piece of information we are given in the question is that the area of this triangle is 74 root three square centimeters.

Now, a formula that will be useful to us here is the trigonometric formula for the area of a triangle. This tells us that in a triangle π΄π΅πΆ, where the uppercase letters π΄, π΅, and πΆ represent the measures of the three angles in the triangle and the lowercase letters π, π, and π represent the lengths of the three opposite sides to these angles, then the area of this triangle is equal to a half ππ sin πΆ. Now, the letters π, π, and π here arenβt actually that important as long as we remember what they represent. The lowercase letters π and π represent the lengths of any two sides in a triangle and the uppercase letter πΆ represents the measure of their included angle. We can calculate the area of any triangle using this formula, provided we have this specific combination of information.

Returning to our triangle π΄π΅πΆ then, we know the length of one side in this triangle and we want to calculate the length of another side. We also know the measure of the included angle between these two sides and the area of the triangle. We can therefore form an equation using the trigonometric formula. A half multiplied by π΄π΅ multiplied by π΅πΆ multiplied by sin of 60 degrees is equal to 74 root three. The length of π΄π΅ is 18 centimeters, and sin of 60 degrees is root three over two. So, we have a half multiplied by 18 multiplied by π΅πΆ multiplied by root three over two equals 74 root three.

And we can now solve this equation to find the length of π΅πΆ. Simplifying on the left-hand side, we have 18 root three over four multiplied by π΅πΆ, and this is equal to 74 root three. We can then cancel a factor of root three from each side of the equation. And we can also cancel 18 over four down to nine over two by dividing both the numerator and denominator by two. To find π΅πΆ, we need to divide both sides of the equation by nine over two, which is the same as multiplying by the reciprocal of this value, which is two-ninths. We find then that π΅πΆ is equal to 148 over nine.

The question specifies that we should give our answer to two decimal places, so we need to evaluate this as a decimal. Itβs 16.4 recurring, which to two decimal places is 16.44. π΅πΆ is a length, so it will have the same length units as the other values in the question, which were centimeters.

So, by recalling the trigonometric formula for the area of a triangle and then working backwards from knowing the area of this triangle, we found that length π΅πΆ to two decimal places is 16.44 centimeters.