»  Solutions to puzzles in my VDARE.com monthly Diary

  December 2023

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In the Math Corner of my December 2023 Diary I posted the following brainteaser, another one from Southall & Pantaloni's Geometry Snacks: Bite Size Problems & How to Solve Them.

Everything that looks equilateral here, is; everything that looks like a midpoint, is. What fraction of the big triangle is shaded?

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•  Solution

Brainteaser

My solution was to draw a (dotted) line from the right-side midpoint to the base midpoint. Two baby equilateral triangles are now on show. It's plain that together they contain half the area of the main triangle. The other half is a rhombus of which the shaded area constitutes a half; so it'a a quarter of the main triangle.

Several readers used the formula half-base-times-height for the area of any triangle. Suppose the main equilateral triangle has side length two units and height h. (The height is actually √3, but you don't even need to know that.)

Now the area of the main triangle, half base times height, is h. The shaded area has "base" (top side) length 1, height half h, so area one-quarter h.

Either way, if you were fooling with sines and cosines, you were overthinking it.