Rectangle 𝐴𝐵𝐶𝐷 has 𝐴𝐵 equals 25 and 𝐵𝐶 equals 36. Draw two perpendiculars segment 𝐵𝐻 and segment 𝐴𝑂 to the plane of the rectangle 𝐴𝐵𝐶𝐷 in the same direction, such that segment 𝐵𝐻 and segment 𝐴𝑂 are both of length 27. What is the area of 𝐶𝐷𝑂𝐻?
Based on the fact that these line segments are perpendicular to the plane of our rectangle, we know that we’ll be operating in three dimensions. If we let the rectangle 𝐴𝐵𝐶𝐷 be part of this 𝑥𝑦-plane, then the perpendiculars 𝐵𝐻 and 𝐴𝑂 will extend upward in the 𝑧-direction perpendicular to the 𝑥𝑦-plane. Both the perpendiculars measure 27. 𝐴𝐵 measures 25, and 𝐵𝐶 36. We should note here that you could draw many different forms of this diagram. The main purpose of the diagram is to help us visualize these shapes.
Before we calculate the area of 𝐶𝐷𝑂𝐻, let’s see which part of the diagram that would be. 𝐶𝐷𝑂𝐻 is this rectangle on our diagram that I’ve highlighted in pink. To find the area of 𝐶𝐷𝑂𝐻, we need to identify the length and the width. The segment 𝐶𝐷 was part of the original rectangle 𝐴𝐵𝐶𝐷. It’s parallel to line segment 𝐴𝐵 and has the same length, so it’s 25. The width here will be the distance from 𝐶 to 𝐻. Since we know that 𝐵𝐻 is perpendicular to 𝐵𝐶, we can use what we know about right triangles to find the width.
𝐵𝐻 equals 27; 𝐵𝐶 equals 36. We’ll use the Pythagorean theorem, which says 𝑐 squared equals 𝑎 squared plus 𝑏 squared, where 𝑐 is the hypotenuse of a right triangle and 𝑎 and 𝑏 are the other two sides. Our missing side — we’ll call 𝑤 squared — is equal to 27 squared plus 36 squared, which equals 2025. Taking the square root of both sides, we find that 𝑤 equals 45. If we plug that back into our original diagram, we’ll see that the rectangle 𝐶𝐷𝑂𝐻 has a length of 25 and a width of 45.
To find the area of a rectangle, we multiply the length by the width. When we do that, we find that the area equals 1125. We weren’t given any units, so we can just identify this as units squared. The area of 𝐶𝐷𝑂𝐻 is 1125 units squared.