Video: Finding Missing Side Lengths in a Composite Figure Using Ratios

𝐴𝐡𝐢𝐷 is a rectangle in which 𝐴𝐡 = 18 cm and 𝐢𝐻𝐸𝐹 is a square whose side length is 10 cm, where 𝐢𝐻/𝐻𝐡 = 5/9. Find the length of segment 𝐴𝐷 and the perimeter of the shaded part.

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Video Transcript

𝐴𝐡𝐢𝐷 is a rectangle in which 𝐴𝐡 equals 18 centimetres and 𝐢𝐻𝐸𝐹 is a square whose side length is 10 centimetres, where 𝐢𝐻 to 𝐻𝐡 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 𝐴𝐡𝐢𝐷 is a rectangle and the length of 𝐴𝐡 is 18 centimetres. We are also told that 𝐢𝐻𝐸𝐹 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: 𝐢𝐻 to 𝐻𝐡 is equal to five to nine.

So let’s begin by finding the length of 𝐴𝐷. We know that 𝐴𝐡𝐢𝐷 is a rectangle. So the length of 𝐴𝐷 will be equal to the length of 𝐡𝐢 because opposite sides are equal on a rectangle. However, we don’t know the length of 𝐡𝐢. However, we know a piece of 𝐡𝐢. We know that 𝐻𝐢 or 𝐢𝐻 is 10 centimetres and we have this proportion.

So we can replace 𝐢𝐻 with 10. And then, it will allow us to find 𝐻𝐡. And that will be very helpful because 𝐢𝐻 plus 𝐻𝐡 will be the entire side length of 𝐡𝐢, 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 𝐻𝐡 is five 𝐻𝐡 and 10 times nine is equal to 90. So to solve for 𝐻𝐡, we divide both sides of the equation by five. And we find that the length of 𝐻𝐡 is 18. And we can use this to find the length of 𝐡𝐢.

We said that the length of 𝐡𝐢 would be the length of 𝐢𝐻 plus the length of 𝐻𝐡, so 10 centimetres plus 18 centimetres. So 𝐡𝐢 is equal to 28 centimetres. so if 𝐡𝐢 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 𝐴𝐡 plus the length of 𝐡𝐻 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 𝐴𝐡 is equal to 18 centimetres, which is shown here. We also know the length of 𝐡𝐻 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 𝐢𝐹, 10 centimetres, plus 𝐹𝐷, the one that we don’t know, will be the entire side length of the rectangle, which is the side 𝐢𝐷. So if 𝐴 to 𝐡, that side length, is 18 centimetres, then 𝐢 to 𝐷 will be 18 centimetres. And we already know the length of 𝐢𝐹; 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.

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