WEBVTT
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Rectangle π΄π΅πΆπ· has π΄π΅ equals 25 and π΅πΆ equals 36.
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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.
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What is the area of πΆπ·ππ»?
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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.
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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.
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Both the perpendiculars measure 27.
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π΄π΅ measures 25, and π΅πΆ 36.
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We should note here that you could draw many different forms of this diagram.
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The main purpose of the diagram is to help us visualize these shapes.
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Before we calculate the area of πΆπ·ππ», letβs see which part of the diagram that would be.
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πΆπ·ππ» is this rectangle on our diagram that Iβve highlighted in pink.
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To find the area of πΆπ·ππ», we need to identify the length and the width.
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The segment πΆπ· was part of the original rectangle π΄π΅πΆπ·.
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Itβs parallel to line segment π΄π΅ and has the same length, so itβs 25.
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The width here will be the distance from πΆ to π».
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Since we know that π΅π» is perpendicular to π΅πΆ, we can use what we know about right triangles to find the width.
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π΅π» equals 27; π΅πΆ equals 36.
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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.
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Our missing side β weβll call π€ squared β is equal to 27 squared plus 36 squared, which equals 2025.
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Taking the square root of both sides, we find that π€ equals 45.
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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.
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To find the area of a rectangle, we multiply the length by the width.
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When we do that, we find that the area equals 1125.
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We werenβt given any units, so we can just identify this as units squared.
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The area of πΆπ·ππ» is 1125 units squared.