Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.6 AIME, harder Prove it

8. Prove that the area of the projection of a polygon located in the plane $\alpha$ onto the plane $\beta$ is equal to $S \cos \varphi$ , where $S$ is the area of the polygon, and $\varphi$ is the angle between the planes $\alpha$ and $\beta$ .

Solution

8. The statement of the problem is obvious for a triangle, one side of which lies on the line of intersection of planes $\alpha$ and $\beta$ . Then, it can be proven to be valid for an arbitrary triangle, and subsequently for an arbitrary polygon.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.