Question Video: Finding the Perimeter of a Triangle with Sides Given as Algebraic Expressions

Video Transcript

In the figure, find the values of 𝑥 and 𝑦.

In this question, we have a triangle 𝐴𝐵𝐶. And the lengths of the sides are given either as numerical values or as algebraic expressions. We need to work out what the values of 𝑥 and 𝑦 are. But let’s begin by figuring out what type of triangle 𝐴𝐵𝐶 is. We’re given that this angle 𝐵𝐴𝐶 is 60 degrees and the other angle 𝐴𝐶𝐵 is also 60 degrees.

We may be able at this point to work out what the angle 𝐴𝐵𝐶 is, but let’s step through it. We can remember that the angles in a triangle add up to 180 degrees. So we could work out angle 𝐴𝐵𝐶 by having 180 degrees subtract 60 degrees subtract 60 degrees, which leaves us with 60 degrees. So now we have a triangle that has three equal interior angles. This means that we can say that triangle 𝐴𝐵𝐶 is an equilateral triangle.

Importantly for us, we now know that this means that all sides in the triangle will be the same length. We could therefore say that 𝐴𝐵 is equal to 𝐴𝐶, which means that 𝑥 plus six must be equal to 26. We could now solve this to find the value of 𝑥. Subtracting six from both sides of this equation would give us that 𝑥 is equal to 20. So now we’ve found our first value.

Let’s see if we can set up a similar equation to find the value of 𝑦. We can write that 𝐵𝐶 is equal to the length 𝐴𝐶. It’s also of course equal to 𝐴𝐵 as well. 𝐵𝐶 is of length 𝑦 plus three, and 𝐴𝐶 is of length 26. So we can set these equal and solve to find 𝑦. This time, we’ll subtract three from both sides, to give us that 𝑦 must be equal to 23. We could then give our answers to the question that 𝑥 is equal to 20 and 𝑦 is equal to 23.