Question Video: Using the Congruence between Two Polygons and the Perimeter of One to Solve a Problem Mathematics • First Year of Preparatory School

Video Transcript

The perimeter of the polygon 𝐴𝐵𝐶𝐷𝐸 is 176 centimeters and 𝐴𝐵𝐶𝐷𝐸 is congruent to 𝐹𝑀𝐿𝐷𝐸. Given that 𝐸 is on the line 𝐴𝐹 and 𝐷𝐸 equals 48 centimeters, find the perimeter of the figure 𝐴𝐵𝐶𝐷𝐿𝑀𝐹.

In this problem, we are asked to find the perimeter of the entire figure, that is, the polygon 𝐴𝐵𝐶𝐷𝐿𝑀𝐹. We are given the fact that the perimeter of the shape on the right, that’s 𝐴𝐵𝐶𝐷𝐸, is 176 centimeters and that the length of 𝐷𝐸 is 48 centimeters. However, knowing these facts alone wouldn’t help us to work out the perimeter of the larger polygon. We will therefore also need to use the information that 𝐴𝐵𝐶𝐷𝐸 is congruent to 𝐹𝑀𝐿𝐷𝐸, which is the polygon on the left.

We can recall that congruent polygons have all pairs of corresponding angles congruent and all pairs of corresponding sides congruent. So the polygons 𝐴𝐵𝐶𝐷𝐸 and 𝐹𝑀𝐿𝐷𝐸 will be exactly the same shape and size. And the congruency relationship indicates the congruent sides. For example, the line segment 𝐴𝐵 is congruent to the line segment 𝐹𝑀. And the line segment 𝐶𝐷 is congruent to line segment 𝐿𝐷.

And so given that we know the perimeter of 𝐴𝐵𝐶𝐷𝐸 is 176 centimeters, then even if we don’t know the length of every individual line segment, we know that if we started at vertex 𝐴 and traveled to vertex 𝐵 then to vertex 𝐶 then to 𝐷 to 𝐸 and back to 𝐴, the total distance traveled would be 176 centimeters. And because congruent polygons have the same side lengths, then this will be the same as the perimeter of polygon 𝐹𝑀𝐿𝐷𝐸. That’s because when we travel from vertex 𝐹 to vertex 𝑀, this is the same length as the line segment 𝐴𝐵. And all the corresponding lengths will be the same as we journey from vertex 𝑀 to 𝐿 to 𝐷 to 𝐸 and back to 𝐹. So the perimeter of 𝐹𝑀𝐿𝐷𝐸 is also 176 centimeters.

Now, a very common mistake to make here when finding the perimeter of the shape 𝐴𝐵𝐶𝐷𝐿𝑀𝐹 would be to simply add the two perimeters together. But remember, both the perimeters included the line segment 𝐷𝐸. And this wouldn’t be included in the perimeter of the polygon 𝐴𝐵𝐶𝐷𝐿𝑀𝐹. And if we add the perimeters, it would be included twice.

So, to find the perimeter of 𝐴𝐵𝐶𝐷𝐿𝑀𝐹, we do add the two perimeters together. But we must also subtract two times the length of line segment 𝐷𝐸. And given that 𝐷𝐸 is 48 centimeters, we have that the perimeter is equal to 176 plus 176 minus two times 48, which can be simplified to give us the answer that the perimeter of 𝐴𝐵𝐶𝐷𝐿𝑀𝐹 is 256 centimeters.