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.
