Video: Finding the Area of a Rectangle given Its Dimensions

Rectangle 𝐴𝐡𝐢𝐷 has 𝐴𝐡 = 25 and 𝐡𝐢 = 36. Suppose perpendiculars 𝐡𝐻 and 𝐴𝑂 are both of length 27. What is the area of 𝐢𝐷𝑂𝐻?

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Video Transcript

Rectangle 𝐴𝐡𝐢𝐷 has 𝐴𝐡 equals 25 and 𝐡𝐢 equals 36. Suppose perpendiculars 𝐡𝐻 and 𝐴𝑂 are both of length 27. What is the area of 𝐢𝐷𝑂𝐻?

The first thing to do in a question like this is to sketch a diagram. We’re told that we have a rectangle. And coming from this will be two perpendiculars. Therefore, we’ll be working in three dimensions. So it might be useful to draw our rectangle like this, ready to add in the perpendiculars. Our first perpendicular starts at point 𝐡 and goes up to a point 𝐻. Our second perpendicular starts at 𝐴 and goes up to a point 𝑂. We’re told that these perpendiculars are both of length 27. So our perpendiculars should be roughly the same length. We can then add in the dimensions to our diagram.

It’s worth noting that there are a number of different diagrams we could’ve drawn. What we’re really looking for is something to help us with our calculations. Here, we’re asked to find the area of 𝐢𝐷𝑂𝐻. 𝐢𝐷𝑂𝐻 would look like this on our diagram, and it would be a rectangle. In order to find out the area of this rectangle, 𝐢𝐷𝑂𝐻, we would need to know the length and the width, which we could multiply together to find the area. The width here would be equal to the line 𝐴𝐡, which means it will be 25 units long. What we do need to work out here is the length, the line 𝑂𝐷.

As we know that 𝑂𝐴 is a perpendicular, that means that we have a right triangle in 𝑂𝐴𝐷. We can therefore apply the Pythagorean theorem, which tells us that the square on the hypotenuse is equal to the sum of the squares on the other two sides. It’s often helpful to draw out the triangles that we’re going to use. We know that the length 𝑂𝐴 is 27 and the length 𝐴𝐷 will be the same as the length 𝐡𝐢, 36. We want to find the length 𝑂𝐷, which we can define as any length, but here we can call it π‘₯. Filling in the values into the Pythagorean theorem, we have the hypotenuse as π‘₯ and the two shorter sides of 27 and 36. And it doesn’t matter which way round we write those. We can evaluate our calculation as π‘₯ squared equals 1296 plus 729. So π‘₯ squared equals 2025. To find π‘₯, we take the square root of both sides, so π‘₯ equals the square root of 2025.

Usually, we keep our answer in this square root form. But actually, 2025 is a square number. So π‘₯ would be 45 units. Now, we’ve calculated this length 𝑂𝐷 as 45. We can go ahead and work out the area. Multiplying our values 45 and 25, which give us that the area of 𝐢𝐷𝑂𝐻 is equal to 1125. We weren’t given any units in the question, but if we needed to give units here, they would of course be square units for the area.

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