If triangle 𝐴𝐵𝐶 is congruent to triangle 𝐴𝐵𝐷, the perimeter of 𝐴𝐶𝐵𝐷 equals 394 centimetres, and 𝐴𝐵 equals 56 centimetres, find the perimeter of triangle 𝐴𝐵𝐶.
So, in this diagram, we can see two triangles, 𝐴𝐵𝐶 and 𝐴𝐵𝐷. The symbol in the question means congruent. And congruent means the same shape and size. So, the corresponding sides in triangles 𝐴𝐵𝐶 and 𝐴𝐵𝐷 are the same length, and the corresponding angles would be the same size. We’re told that the perimeter of a 𝐴𝐶𝐵𝐷 is 394 centimetres. And we can recall that the perimeter of a shape is the total distance around the outside edges. And in this case, our perimeter is of 𝐴𝐶𝐵𝐷, so that’s our quadrilateral and not one of the triangles.
So, to answer the question, we’re asked to find the perimeter of triangle 𝐴𝐵𝐶. So, let’s return to the information that the perimeter of the whole quadrilateral is 394 centimetres. That means if we were to walk along the outside edge, the total length we would walk would be 394 centimetres. But let’s say then we only wanted to walk from 𝐴 to 𝐶 to 𝐵. Because our triangles are congruent, we can say that our length 𝐴𝐶 is equivalent to the length 𝐴𝐷. And we also know that there are two more equal lengths. Our line 𝐶𝐵 is equal to our line 𝐵𝐷. So, we could say that walking from 𝐴 to 𝐶 and then from 𝐶 to 𝐵 would be half of our perimeter, which would be half of 394 centimetres.
Let’s see how we can write this formally in a more mathematical way. We can call our length 𝐴𝐶 𝑥. And because we know that the length 𝐴𝐷 is the same length, we can also say that this would be equal to 𝑥. We can use the letter 𝑦 to signify our length 𝐶𝐵. And since 𝐵𝐷 is also the same length, we can also call this 𝑦. So, now, to represent the perimeter of 𝐴𝐶𝐵𝐷, we would add the lengths around the outside, giving us 𝑥 plus 𝑦 plus 𝑦 plus 𝑥, which we can write as two 𝑥 plus two 𝑦. We can factor our right-hand side as two and then our parentheses 𝑥 plus 𝑦. And then, to work out just 𝑥 plus 𝑦, we need to rearrange our equation.
And we can do that by dividing both sides of our equation by two, which would give us that the perimeter of 𝐴𝐶𝐵𝐷 over two is equal to our 𝑥 plus 𝑦. And since we know that the perimeter of 𝐴𝐶𝐵𝐷 is equal to 394 centimetres, we can substitute that value in, giving us that 𝑥 plus 𝑦 is equal to 394 over two, which is 197 centimetres. So, now, to find the perimeter of triangle 𝐴𝐵𝐶, we know that the sum of the two lengths 𝐴𝐶 and 𝐶𝐵 is 197 centimetres. So, we need to find the final length 𝐴𝐵 and add it on.
And we’re given this in the question. The light 𝐴𝐵 equals 56 centimetres. So, to find the perimeter, we add the three lengths in. We know that our length 𝐴𝐶 was signified by 𝑥, our length 𝐶𝐵 which was signified by 𝑦, and our length 𝐴𝐵. So, using our values then, this will be equivalent to 197 plus 56. Simplifying this will give us 253 and the units of centimetres since that’s our length units. So, the final answer for the perimeter of our triangle 𝐴𝐵𝐶 is 253 centimetres.