One side of an equilateral triangle has length four 𝑥 plus seven. Another side has length three 𝑥 plus eight. Find the perimeter of the triangle.
So in this question, we’ve been given algebraic expressions for the length of two sides of an equilateral triangle and asked to find its perimeter. In order to do so, we need to know the lengths of the sides and therefore the value of the letter 𝑥. The key factor we need to use in this question is this: the three sides of an equilateral triangle are equal in length.
Therefore, we can form an equation by setting the expressions that we’re given for the lengths of two other sides equal to one another. We have four 𝑥 plus seven is equal to three 𝑥 plus eight. We’ll now solve this equation to find the value of 𝑥.
The first step is to subtract seven from each side. Doing so gives four 𝑥 is equal to three 𝑥 plus one. The next step is to subtract three 𝑥 from both sides so that all the 𝑥s are on the left-hand side of the equation. This in fact solves the equation and gives 𝑥 is equal to one. So we found the value of 𝑥. The reason for doing this was so that we can calculate the length of each side of the triangle.
We can substitute this value of 𝑥 into either of the two expressions: either four 𝑥 plus seven or three 𝑥 plus eight. And they both give the same result. I’ve chosen to substitute into the expression four 𝑥 plus seven. So I have four multiplied by one plus seven which is four plus seven. And therefore, the side length of this equilateral triangle is 11.
Now, the question asked us not to find just the side length, but the perimeter of the triangle. So we need to find the sum of the three sides. Remember all three sides of this triangle are the same length. And therefore, the perimeter is equal to three multiplied by 11. The perimeter of the triangle is 33 units.