### Video Transcript

Rectangle π΄π΅πΆπ· has π΄π΅ equals 25 and π΅πΆ equals 36. Draw two perpendiculars segment π΅π» and segment π΄π to the plane of the rectangle π΄π΅πΆπ· in the same direction, such that segment π΅π» and segment π΄π are both of length 27. What is the area of πΆπ·ππ»?

Based on the fact that these line segments are perpendicular to the plane of our rectangle, we know that weβll be operating in three dimensions. If we let the rectangle π΄π΅πΆπ· be part of this π₯π¦-plane, then the perpendiculars π΅π» and π΄π will extend upward in the π§-direction perpendicular to the π₯π¦-plane. Both the perpendiculars measure 27. π΄π΅ measures 25, and π΅πΆ 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 πΆπ·ππ», letβs see which part of the diagram that would be. πΆπ·ππ» is this rectangle on our diagram that Iβve highlighted in pink. To find the area of πΆπ·ππ», we need to identify the length and the width. The segment πΆπ· was part of the original rectangle π΄π΅πΆπ·. Itβs parallel to line segment π΄π΅ and has the same length, so itβs 25. The width here will be the distance from πΆ to π». Since we know that π΅π» is perpendicular to π΅πΆ, we can use what we know about right triangles to find the width.

π΅π» equals 27; π΅πΆ equals 36. Weβll use the Pythagorean theorem, which says π squared equals π squared plus π squared, where π is the hypotenuse of a right triangle and π and π are the other two sides. Our missing side β weβll call π€ squared β is equal to 27 squared plus 36 squared, which equals 2025. Taking the square root of both sides, we find that π€ equals 45. If we plug that back into our original diagram, weβll see that the rectangle πΆπ·ππ» 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 werenβt given any units, so we can just identify this as units squared. The area of πΆπ·ππ» is 1125 units squared.