Given that 𝐴𝐵𝐶𝐷 is a parallelogram and 𝐶𝑀 equals 8.6 centimeters, find the perimeter of triangle 𝐴𝐵𝐶.
So here we have our parallelogram 𝐴𝐵𝐶𝐷. And we’re told that 𝐶𝑀 equals 8.6 centimeters. So let’s fill that into the diagram. We’re asked to work out the perimeter of triangle 𝐴𝐵𝐶. We can recall that the perimeter of a shape is the sum of the outside edge lengths. In other words, it’s the distance around the outside edges. So to work out the perimeter of our orange triangle, triangle 𝐴𝐵𝐶, we can see that we are missing a few lengths. We’re missing the length 𝐴𝑀 and the length 𝐵𝐶.
Let’s see if we can use any of the facts that we know about parallelograms to help us find these missing lengths. We can remember that opposite sides of a parallelogram are parallel and congruent. So now we know that our opposite sides are the same lengths. We know that our side length 𝐶𝐷 is 10 centimeters and also that our side 𝐵𝐶 must be 11 centimeters. So now, if we look at our length 𝐴𝑀, which is part of the diagonal 𝐴𝐶, we can use the fact that the diagonals of a parallelogram bisect each other. As they bisect each other, this means that the two parts of each diagonal are the same length. So our length 𝐷𝑀 is equal to our length 𝑀𝐵. And importantly for us here, we now know that the length 𝐶𝑀 is equal to the length 𝐴𝑀, giving us that our final missing length is 8.6 centimeters.
So to find our perimeter of triangle 𝐴𝐵𝐶, we add these lengths together. Which is 8.6 plus 8.6 plus 10 plus 11, simplifying to 38.2 centimeters. Which is our final answer for the perimeter of triangle 𝐴𝐵𝐶.