𝐴𝐵𝐶𝐷 is a rectangle in which 𝐴𝐵 equals 18 centimetres and 𝐶𝐻𝐸𝐹 is a square whose side length is 10 centimetres, where 𝐶𝐻 to 𝐻𝐵 is equal to five to nine. Find the length of segment 𝐴𝐷 and the perimeter of the shaded part.
Let’s begin by going through the information that’s given. We’re told that 𝐴𝐵𝐶𝐷 is a rectangle and the length of 𝐴𝐵 is 18 centimetres. We are also told that 𝐶𝐻𝐸𝐹 is a square, whose side length is 10 centimetres. And with a square, all sides are equal. So they’re all labeled 10 centimetres, unless they were given this relationship, this proportion: 𝐶𝐻 to 𝐻𝐵 is equal to five to nine.
So let’s begin by finding the length of 𝐴𝐷. We know that 𝐴𝐵𝐶𝐷 is a rectangle. So the length of 𝐴𝐷 will be equal to the length of 𝐵𝐶 because opposite sides are equal on a rectangle. However, we don’t know the length of 𝐵𝐶. However, we know a piece of 𝐵𝐶. We know that 𝐻𝐶 or 𝐶𝐻 is 10 centimetres and we have this proportion.
So we can replace 𝐶𝐻 with 10. And then, it will allow us to find 𝐻𝐵. And that will be very helpful because 𝐶𝐻 plus 𝐻𝐵 will be the entire side length of 𝐵𝐶, which we said would be equal to the length of 𝐴𝐷, what we were asked to find.
So to solve this proportion, let’s cross multiply. Five times 𝐻𝐵 is five 𝐻𝐵 and 10 times nine is equal to 90. So to solve for 𝐻𝐵, we divide both sides of the equation by five. And we find that the length of 𝐻𝐵 is 18. And we can use this to find the length of 𝐵𝐶.
We said that the length of 𝐵𝐶 would be the length of 𝐶𝐻 plus the length of 𝐻𝐵, so 10 centimetres plus 18 centimetres. So 𝐵𝐶 is equal to 28 centimetres. so if 𝐵𝐶 is 28 centimetres, 𝐴𝐷 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 𝐴𝐵 plus the length of 𝐵𝐻 plus the length of 𝐻𝐸 plus the length of 𝐸𝐹 plus the length of 𝐹𝐷 plus the length from 𝐷 to 𝐴.
Now, we already know a few of these. We know that 𝐴𝐵 is equal to 18 centimetres, which is shown here. We also know the length of 𝐵𝐻 is equal to 18 centimetres, the length of 𝐻𝐸 is 10 centimetres, the length of 𝐸𝐹 is 10 centimetres. We don’t know the length of 𝐹𝐷, but we do now the length of 𝐷𝐴; it’s 28 centimetres.
So we need to solve for the length of 𝐹𝐷. We do know that 𝐶𝐹, 10 centimetres, plus 𝐹𝐷, the one that we don’t know, will be the entire side length of the rectangle, which is the side 𝐶𝐷. So if 𝐴 to 𝐵, that side length, is 18 centimetres, then 𝐶 to 𝐷 will be 18 centimetres. And we already know the length of 𝐶𝐹; it’s 10. So then we can solve for the length of 𝐹𝐷, our missing piece for the perimeter.
So we subtract 10 from both sides and find that the length of 𝐹𝐷 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 𝐴𝐷 is 28 centimetres and the perimeter of the shaded part would be 92 centimetres.