Question 27

Consider a square ABCD with midpoints E, F, G and H of sides AB, BC, CD and DA. Let L denote the line passing through F and H. Consider points P and Q on the line L inside the square such that the angles APD and BQC are both equal 120 degrees. What is the ratio ABQCDP to the remaining area of ABCD?

Solution

We can make the following figure from the information given in the question:

Suppose the side of the square ABCD = 2a

Angle HAP= 30$$^{\circ\ }$$

HA= a

HP=$$\frac{a}{\sqrt{\ 3}}$$

Area HAP = $$\frac{a^2}{2\sqrt{\ 3}}$$

Remaining Area = 4* $$\frac{a^2}{2\sqrt{\ 3}}$$

Area of the shaded region= 4a$$^2$$ - 4* $$\frac{a^2}{2\sqrt{\ 3}}$$

Ratio= Shaded/ Remaining = 2$$\sqrt{\ 3}$$-1

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