### Video Transcript

If triangle π΄π΅πΆ is congruent to triangle π΄π΅π·, the perimeter of π΄πΆπ΅π· equals 394 centimetres, and π΄π΅ equals 56 centimetres, find the perimeter of triangle π΄π΅πΆ.

So, in this diagram, we can see two triangles, π΄π΅πΆ and π΄π΅π·. The symbol in the question means congruent. And congruent means the same shape and size. So, the corresponding sides in triangles π΄π΅πΆ and π΄π΅π· are the same length, and the corresponding angles would be the same size. Weβre told that the perimeter of a π΄πΆπ΅π· 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 π΄πΆπ΅π·, so thatβs our quadrilateral and not one of the triangles.

So, to answer the question, weβre asked to find the perimeter of triangle π΄π΅πΆ. So, letβs 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 letβs say then we only wanted to walk from π΄ to πΆ to π΅. Because our triangles are congruent, we can say that our length π΄πΆ is equivalent to the length π΄π·. And we also know that there are two more equal lengths. Our line πΆπ΅ is equal to our line π΅π·. So, we could say that walking from π΄ to πΆ and then from πΆ to π΅ would be half of our perimeter, which would be half of 394 centimetres.

Letβs see how we can write this formally in a more mathematical way. We can call our length π΄πΆ π₯. And because we know that the length π΄π· is the same length, we can also say that this would be equal to π₯. We can use the letter π¦ to signify our length πΆπ΅. And since π΅π· is also the same length, we can also call this π¦. So, now, to represent the perimeter of π΄πΆπ΅π·, we would add the lengths around the outside, giving us π₯ plus π¦ plus π¦ plus π₯, which we can write as two π₯ plus two π¦. We can factor our right-hand side as two and then our parentheses π₯ plus π¦. And then, to work out just π₯ plus π¦, 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 π΄πΆπ΅π· over two is equal to our π₯ plus π¦. And since we know that the perimeter of π΄πΆπ΅π· is equal to 394 centimetres, we can substitute that value in, giving us that π₯ plus π¦ is equal to 394 over two, which is 197 centimetres. So, now, to find the perimeter of triangle π΄π΅πΆ, we know that the sum of the two lengths π΄πΆ and πΆπ΅ is 197 centimetres. So, we need to find the final length π΄π΅ and add it on.

And weβre given this in the question. The light π΄π΅ equals 56 centimetres. So, to find the perimeter, we add the three lengths in. We know that our length π΄πΆ was signified by π₯, our length πΆπ΅ which was signified by π¦, and our length π΄π΅. 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 thatβs our length units. So, the final answer for the perimeter of our triangle π΄π΅πΆ is 253 centimetres.