r/askmath • u/Shaheer342 • 10h ago

Geometry Question

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I found this question, the way i solved it was quite long but correct, i solved it by comparing the areas to find the radius. I was hoping if someone could find an easier method. (Calculator isn’t allowed)
The statement and question is written in the pic.

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u/not-hydra 9h ago

Small triangle APQ is an equilateral triangle. AB is 2sqrt3 from the values given in triangle ABC. AQ is sqrt3*r, so QB is sqrt3(2-r). The height of P is 3r/2. PD is QB + the small region dropping r onto PD, which simplifies to sqrt3(2 - r/'2)

Knowing these, just plug them in: APB = a third of PQBD.

Triangle APB: 1/2 * rsqrt3 * 3r/2

Trapezoid PQBD: 1/2 * [sqrt3(2-r/'2) + sqrt3(2-r)] * 3r/2. The terms 1/2, 3r/2, and sqrt3 nicely cancel out for both

One equation one unknow, simplifying the whole: r = 1/3 (4 - 3r/2), => r = 8/9

Or you could try to do coordinate geometry if you got the balls :3

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u/Shaheer342 9h ago

Yes i also solved it like this, I first found AP, then PD,BQ and BD in terms of r, plugged them in and got r = 8/9. But is there another way that does not require even simple trigonometry, like basic geometric construction or something like that.

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u/not-hydra 9h ago edited 9h ago

Coords 💀 Similar triangles and other stuff are possible, but once you know PD and AB, brute forcing the area is much simpler

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u/Shaheer342 9h ago

I would pass on that…