Given that triangle 𝐴𝐵𝐶 is similar to triangle 𝐴 prime 𝐵 prime 𝐶 prime, determine the perimeter of triangle 𝐴 prime 𝐵 prime 𝐶 prime.
We’ve been given that these two triangles are similar to one another. And we can recall that similar polygons have two key properties. Firstly, corresponding angles are congruent. And secondly, corresponding sides are proportional. We can use the ordering of letters in the similarity statement to determine which vertices on the two polygons correspond to one another.
Vertex 𝐴 on the first triangle corresponds to vertex 𝐴 prime on the second, as we might expect. Vertex 𝐵 corresponds to vertex 𝐵 prime. And finally, vertex 𝐶 corresponds to vertex 𝐶 prime. This also helps us to determine which sides on the two triangles are corresponding to one another. And we may find it helpful to highlight corresponding sides in the same color.
Now, we’ve been asked to determine the perimeter of triangle 𝐴 prime 𝐵 prime 𝐶 prime. And in order to do this, we first need to determine its three side lengths. We’re given that the length of side 𝐴 prime 𝐶 prime is two units. The length of side 𝐴 prime 𝐵 prime is expressed in terms of an unknown 𝑥. And we currently have no value or expression for the length of side 𝐵 prime 𝐶 prime. However, because we know that corresponding sides on the two similar triangles are proportional, we can use the values and expressions we have been given together with the values and expressions we’ve been given for the lengths of the sides of triangle 𝐴𝐵𝐶 to form an equation.
Using the lengths of 𝐴𝐶, 𝐴 prime 𝐶 prime, 𝐴𝐵, and 𝐴 prime 𝐵 prime, we have the equation 𝑥 plus three over two is equal to five over 𝑥. We can now solve this equation to determine the value of 𝑥. And we’ll begin by cross multiplying, to give 𝑥 multiplied by 𝑥 plus three is equal to two multiplied by five, which is 10. Distributing the parentheses on the left-hand side and then subtracting 10 from each side of the equation gives 𝑥 squared plus three 𝑥 minus 10 is equal to zero.
Now, this is a quadratic equation in 𝑥, and it can be solved by factoring. The factored form of this quadratic is 𝑥 plus five multiplied by 𝑥 minus two is equal to zero. And this can be confirmed by redistributing the parentheses. We found these factors by identifying that the first term in each set of parentheses will be 𝑥 so that when we multiply, we get 𝑥 squared. And then, we complete each set of parentheses by looking for two numbers whose sum is the coefficient of 𝑥 — that’s positive three — and whose product is the constant term of negative 10. Those numbers are positive five and negative two.
Next, we recall that if a product is equal to zero, then at least one of the individual factors must itself be equal to zero. So, either 𝑥 plus five equals zero or 𝑥 minus two equals zero. These linear equations can each be solved in one step. To solve the first equation, we subtract five from each side, giving 𝑥 equals negative five. And to solve the second, we add two to each side, giving 𝑥 equals two.
Now, while these are both valid solutions to this quadratic equation, they aren’t both valid solutions in the context of this problem. 𝑥 represents the length of the side 𝐴 prime 𝐵 prime. And so its value must be positive. For this reason, we can disregard the solution 𝑥 equals negative five. And we’ll proceed with only 𝑥 is equal to two. So, we found the length of side 𝐴 prime 𝐵 prime. It was 𝑥 units, which we now know is two units.
Before we can calculate the perimeter of triangle 𝐴 prime 𝐵 prime 𝐶 prime, we need to calculate the length of the final side, side 𝐵 prime 𝐶 prime. We can form another equation, this time using sides 𝐵 prime 𝐶 prime, 𝐵𝐶, 𝐴 prime 𝐵 prime, and 𝐴𝐵. Substituting eight for the length of 𝐵𝐶, two for the length of 𝐴 prime 𝐵 prime, and five for the length of 𝐴𝐵, we have 𝐵 prime 𝐶 prime over eight equals two over five. Multiplying both sides of this equation by eight, we find that 𝐵 prime 𝐶 prime is equal to two times eight over five. That’s 16 over five or, as a decimal, 3.2.
So, using the proportionality of corresponding side lengths on the two similar triangles, we’ve now found the lengths of all three sides of triangle 𝐴 prime 𝐵 prime 𝐶 prime. And so we can calculate its perimeter by summing these three values. We have that the perimeter of triangle 𝐴 prime 𝐵 prime 𝐶 prime is equal to two plus two plus 3.2, which is 7.2. There were no units given for the lengths in the question, so we’re just assuming general length units here. And we have that the perimeter of triangle 𝐴 prime 𝐵 prime 𝐶 prime is 7.2 units.