Triangle Geometry Practice
February 8, 2007
Here are a couple of problems I saved a while back from the Drexel University Math Forum:
1. A square with side length s is inscribed in an equilateral triangle of side length t. t = ks. Find k, expressed as a fraction in simplest radical form.
2. A regular hexagon and an equilateral triangle are both inscribed in the same circle so that the hexagon and the triangle share three vertices. The radius of the circle is 8 units. What is the area of the region between the two polygons?
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Ramblings of a Math Mom 
For the first, I started with an equilateral triangle with side S, and dropped a vertical of length S to make a 30-60-90 triangle. Finding the hypotenuse was just algebra. And then taking care to keep the parts straight and offer the correct ratio.
Jonathan
I’m not following — you dropped a vertical in the equilateral triangle with side S? Then it wouldn’t have length S. I used the little 30-60-90’s already present on the sides at the bottom, with legs S and 1/2(T-S) and hypotenuse (T-S) and then, as you said, it’s all algebra.
I started with a 60-60-60 with side S (in other words, the little triangle on top)
If we labeled the endpoints of the various segments A,B,C,D,E,F,G so that the A is on top and the whole thing is ACF:
I started with ABG (all sides are S) and added BD (length S), BC and CD (lengths derived from 30-60-90 ratios).
Jonathan
Oh, I think that’s the same as what I did. It’s just that BD is already there, so your description confused me.
Difference equals = 3Tan(30)r^2/cos(30)-1.5cos(30)r^2 or about 45 square units in this case.
Correction for problem number 2, the answer is 3/4*3^0.5*radius^2.
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