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Geometry- Hexagon

Problem Solving — algebra and arithmetic
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by s91arvindh » Mon Jun 02, 2014 10:22 pm
ABCDEF is a regular hexagon. P, Q and R are mid-points of AB, CD and EF respectively. Ratio of areas of triangle PQR to hexagon is

a) 1 : 2 b) 1 : 3 c) 2 : 3 d) 4 : 9 e) 3 : 8

Answer e

Source of this question is from a handout received in class.
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by [email protected] » Tue Jun 03, 2014 12:12 am
Hi s91arvindh,

Hexagon's are exceptionally rare on Test Day, so this question is not likely to show on the actual exam.

Before offering a solution, I'm going to give you some hints and let you "play" with this question a bit:

1) A regular hexagon is a 6-sided shape with 720 degrees.
2) Each of the angles in a regular hexagon is 120 degrees.
3) Put a dot in the middle of the hexagon and draw lines to each of the vertices and you'll have 6 identical equilateral triangles.
4) This question is perfect for TESTing VALUES. Try picking a value for a side length and see what you can figure out from there.

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by s91arvindh » Tue Jun 03, 2014 8:30 pm
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Hi Rich,

Thanks a lot for giving a direction to solve this question,

I tried your method with the values of angles,

The area of triangle in the hexagon will be 5*1/2*b*h
I have considered Side as 4, The height of the triangle will be 2squareroot(3)
Area of triangles in hexagon will be 5*1/2*4*2squareroot(3)
Stuck after this..

But when I drawn midpoints as per the question.
There was another hexagon inscribed ( Drawn in brown color)

So can we use the inscribed hexagon as hint for solving this question.

Thanks
s91arvindh
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