# Question Video: Finding Missing Side Lengths in a Composite Figure Using Ratios Mathematics • 6th Grade

π΄π΅πΆπ· is a rectangle in which π΄π΅ = 18 cm and πΆπ»πΈπΉ is a square whose side length is 10 cm, where πΆπ»/π»π΅ = 5/9. Find the length of segment π΄π· and the perimeter of the shaded part.

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### Video Transcript

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

Letβs begin by going through the information thatβs given. Weβre told that π΄π΅πΆπ· is a rectangle and the length of π΄π΅ is 18 centimetres. We are also told that πΆπ»πΈπΉ is a square, whose side length is 10 centimetres. And with a square, all sides are equal. So theyβre all labeled 10 centimetres, unless they were given this relationship, this proportion: πΆπ» to π»π΅ is equal to five to nine.

So letβs begin by finding the length of π΄π·. We know that π΄π΅πΆπ· is a rectangle. So the length of π΄π· will be equal to the length of π΅πΆ because opposite sides are equal on a rectangle. However, we donβt know the length of π΅πΆ. However, we know a piece of π΅πΆ. We know that π»πΆ or πΆπ» is 10 centimetres and we have this proportion.

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

So to solve this proportion, letβs 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΅πΆ.

We said that the length of π΅πΆ would be the length of πΆπ» plus the length of π»π΅, so 10 centimetres plus 18 centimetres. So π΅πΆ is equal to 28 centimetres. so if π΅πΆ 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»πΈ plus the length of πΈπΉ plus the length of πΉπ· 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»πΈ is 10 centimetres, the length of πΈπΉ is 10 centimetres. We donβt know the length of πΉπ·, but we do now the length of π·π΄; itβs 28 centimetres.

So we need to solve for the length of πΉπ·. We do know that πΆπΉ, 10 centimetres, plus πΉπ·, the one that we donβt know, will be the entire side length of the rectangle, which is the side πΆπ·. So if π΄ to π΅, that side length, is 18 centimetres, then πΆ to π· will be 18 centimetres. And we already know the length of πΆπΉ; itβs 10. So then we can solve for the length of πΉπ·, our missing piece for the perimeter.

So we subtract 10 from both sides and find that the length of πΉπ· 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.