𝐴𝐵𝐶 is a triangle where 𝐴𝐵 equals 18 centimeters, the measure of angle 𝐵 equals 60 degrees, and the area of the triangle is 74 root three square centimeters. Find length 𝐵𝐶, giving the answer to two decimal places.
Let’s begin by sketching this triangle as best we can. We know that we have a side of length 18 centimeters, so here’s that side. And then the measure of the angle at 𝐵 is 60 degrees. We can draw in a line that meets the side 𝐴𝐵 at an angle of 60 degrees, but we don’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 𝐵𝐶 could be extended off the screen, and the angle at 𝐴 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 𝐴𝐵𝐶, where the uppercase letters 𝐴, 𝐵, and 𝐶 represent the measures of the three angles in the triangle and the lowercase letters 𝑎, 𝑏, and 𝑐 represent the lengths of the three opposite sides to these angles, then the area of this triangle is equal to a half 𝑎𝑏 sin 𝐶. Now, the letters 𝑎, 𝑏, and 𝑐 here aren’t actually that important as long as we remember what they represent. The lowercase letters 𝑎 and 𝑏 represent the lengths of any two sides in a triangle and the uppercase letter 𝐶 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 𝐴𝐵𝐶 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 𝐴𝐵 multiplied by 𝐵𝐶 multiplied by sin of 60 degrees is equal to 74 root three. The length of 𝐴𝐵 is 18 centimeters, and sin of 60 degrees is root three over two. So, we have a half multiplied by 18 multiplied by 𝐵𝐶 multiplied by root three over two equals 74 root three.
And we can now solve this equation to find the length of 𝐵𝐶. Simplifying on the left-hand side, we have 18 root three over four multiplied by 𝐵𝐶, 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 𝐵𝐶, 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 𝐵𝐶 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. It’s 16.4 recurring, which to two decimal places is 16.44. 𝐵𝐶 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 𝐵𝐶 to two decimal places is 16.44 centimeters.