Video: Finding the Area of a Diagonal Rectangle inside a Rectangular Parallelepiped Given Its Dimensions

𝐴𝐡𝐢𝐷 𝐴′𝐡′𝐢′𝐷′ is a rectangular parallelepiped whose three dimensions are 𝐴𝐡 = 69 cm, 𝐡𝐢 = 55 cm, and 𝐴𝐴′ = 92 cm. Determine the area of the rectangle 𝐢𝐡𝐴′𝐷′.

04:20

Video Transcript

𝐴𝐡𝐢𝐷 𝐴 prime 𝐡 prime 𝐢 prime 𝐷 prime is a rectangular parallelepiped whose three dimensions are 𝐴𝐡 equals 69 centimeters, 𝐡𝐢 equals 55 centimeters, and 𝐴𝐴 prime equals 92 centimeters. Determine the area of the rectangle 𝐢𝐡𝐴 prime 𝐷 prime.

Let’s begin by filling in the given measurements. 𝐴𝐡 is 69 centimeters, 𝐡𝐢 is 55 centimeters, and 𝐴𝐴 prime is 92 centimeters. We’re told that the three-dimensional shape in this figure is a parallelepiped. And we might usually see one drawn like this. However. 𝐴𝐡𝐢𝐷𝐴 prime 𝐡 prime 𝐢 prime 𝐷 prime is described as a rectangular parallelepiped, which is a special type of parallelepiped where all six faces are rectangles. We can alternatively think of this shape as a cuboid or rectangular prism. What it means is that we’ll have right angles here, here, and here.

So let’s see what we’re asked to calculate. It’s the area of the rectangle 𝐢𝐡𝐴 prime 𝐷 prime. This will be the plane that cuts through our solid. We should recall that to find the area of a rectangle, we multiply the length by the width. We know one of the dimensions of this rectangle will be 55 centimeters, but we’ll need to work out this length of 𝐴 prime 𝐡. We’re not given any information as to the length of this line segment 𝐴 prime 𝐡, but let’s consider it as part of this right triangle.

We know that the triangle 𝐴 prime 𝐴𝐡 will be a right triangle because it’s part of this rectangular parallelepiped. As we’re given two lengths in this right triangle and we want to find the length of this third side, then we can apply the Pythagorean theorem. This theorem tells us that the square on the hypotenuse is equal to the sum of the squares on the other two sides. And it’s often written as 𝑐 squared equals π‘Ž squared plus 𝑏 squared, where 𝑐 is the hypotenuse and π‘Ž and 𝑏 are the other two sides.

So let’s say that we define the length of the line segment 𝐴 prime 𝐡 with the letter π‘₯. We can then fill in the given values. The hypotenuse will be π‘₯. That’s the longest side, and it’s always opposite the right angle. And the squares of the other two sides can be written in any order, so we’ll have π‘₯ squared equals 92 squared plus 69 squared. Using a non-calculator method, we can evaluate these squares as 8464 plus 4761, which gives us 13225. Next, we need to take the square root of both sides of this equation in order to find the value of π‘₯.

So π‘₯ is equal to the square root of 13225. And the units for π‘₯ will be centimeters as π‘₯ is a length. Usually at this point when we’re working with the Pythagorean theorem in three dimensions or trigonometry in three dimensions and we have got a value which we haven’t finished using in our calculations, we would say to keep this value in this square root form. However, 13225 is in fact a perfect square, which means that when we find the square root, we get an integer value.

This square root of 13225 is in fact equal to 115. So π‘₯ is 115 centimeters. So now, we have both the length and the width for the rectangle 𝐢𝐡𝐴 prime 𝐷 prime. We can, therefore, find the area by multiplying the length and the width. So we’ll calculate 115 multiplied by 55. This gives us the area of the rectangle 𝐢𝐡𝐴 prime 𝐷 prime of 6325 square centimeters.

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