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Tough and Tricky questions: Geometry.
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If triangle ABD is an equilateral triangle and AB = 6 and CE = 18, what fraction of the trapezoid BACE is shaded?
A. 1/5
B. 1/4
C. 1/3
D. 3/8
E. 1/2
Although Area of Trapezoid formula will make this easier, it is not required. I thought I'd post the way to do this without that formula:
Method Using Only Triangle FormulasSHADED TRIANGLEAB=6, and that triangle is equilateral, so AB=BD=DA=6.
Our height will be 3\(\sqrt{3}\). We can make Right Triangles and determine this. (See picture)
Height = 3\(\sqrt{3}\) for all three trianglesArea of ABD = \(\frac{1}{2}\) * b * h ----> \(\frac{1}{2}\) * 6 * 3\(\sqrt{3}\) ----> 9\(\sqrt{3}\)
OTHER TWO TRIANGLESWe know that CAD=DBE. They are similar triangles. It is given that CE=18, so CD=DE=9.
Base = 9Area of CAD = \(\frac{1}{2}\) * 9 * h
Area of CAD = \(\frac{1}{2}\) * 9 * 3\(\sqrt{3}\) = \((27\sqrt{3})/2\)
We have two of these triangles, so we multiply \((27\sqrt{3})/2\) by 2 and get 27\(\sqrt{3}\)
TOTAL AREA OF ALL THREE TRIANGLES27\(\sqrt{3}\) + 9\(\sqrt{3}\) = 36\(\sqrt{3}\)
FRACTION SHADED9\(\sqrt{3}\) / 36\(\sqrt{3}\) = \(\frac{1}{4}\)
Answer: B
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