### Video Transcript

π΄π΅πΆπ· is a rectangle in which π΄π΅ equals 18 centimetres and πΆπ»πΈπΉ is a square whose side length is 10 centimetres, where πΆπ» to π»π΅ is equal to five to nine. Find the length of segment π΄π· and the perimeter of the shaded part.

Letβs begin by going through the information thatβs given. Weβre told that π΄π΅πΆπ· is a rectangle and the length of π΄π΅ is 18 centimetres. We are also told that πΆπ»πΈπΉ is a square, whose side length is 10 centimetres. And with a square, all sides are equal. So theyβre all labeled 10 centimetres, unless they were given this relationship, this proportion: πΆπ» to π»π΅ is equal to five to nine.

So letβs begin by finding the length of π΄π·. We know that π΄π΅πΆπ· is a rectangle. So the length of π΄π· will be equal to the length of π΅πΆ because opposite sides are equal on a rectangle. However, we donβt know the length of π΅πΆ. However, we know a piece of π΅πΆ. We know that π»πΆ or πΆπ» is 10 centimetres and we have this proportion.

So we can replace πΆπ» with 10. And then, it will allow us to find π»π΅. And that will be very helpful because πΆπ» plus π»π΅ will be the entire side length of π΅πΆ, which we said would be equal to the length of π΄π·, what we were asked to find.

So to solve this proportion, letβs cross multiply. Five times π»π΅ is five π»π΅ and 10 times nine is equal to 90. So to solve for π»π΅, we divide both sides of the equation by five. And we find that the length of π»π΅ is 18. And we can use this to find the length of π΅πΆ.

We said that the length of π΅πΆ would be the length of πΆπ» plus the length of π»π΅, so 10 centimetres plus 18 centimetres. So π΅πΆ is equal to 28 centimetres. so if π΅πΆ is 28 centimetres, π΄π· is equal to 28 centimetres because opposite sides of a rectangle are equal in length.

Lastly, we are asked to find the perimeter of the shaded part. So the perimeter of the shaded part will be the length of π΄π΅ plus the length of π΅π» plus the length of π»πΈ plus the length of πΈπΉ plus the length of πΉπ· plus the length from π· to π΄.

Now, we already know a few of these. We know that π΄π΅ is equal to 18 centimetres, which is shown here. We also know the length of π΅π» is equal to 18 centimetres, the length of π»πΈ is 10 centimetres, the length of πΈπΉ is 10 centimetres. We donβt know the length of πΉπ·, but we do now the length of π·π΄; itβs 28 centimetres.

So we need to solve for the length of πΉπ·. We do know that πΆπΉ, 10 centimetres, plus πΉπ·, the one that we donβt know, will be the entire side length of the rectangle, which is the side πΆπ·. So if π΄ to π΅, that side length, is 18 centimetres, then πΆ to π· will be 18 centimetres. And we already know the length of πΆπΉ; itβs 10. So then we can solve for the length of πΉπ·, our missing piece for the perimeter.

So we subtract 10 from both sides and find that the length of πΉπ· is eight. So we can plug this into our equation. So to find our perimeter, we need to take 18 centimetres plus 18 centimetres plus 10 centimetres plus 10 centimetres plus eight centimetres plus 28 centimetres, resulting in 92 centimetres.

So once again, the length of π΄π· is 28 centimetres and the perimeter of the shaded part would be 92 centimetres.