If \(\triangle{AHE}\) is a right angle triangle and AH || BG || CF and AB=BC=CE=√3, what is the ratio of area of coloured portion to the area of uncoloured portion]?

(A) \(\frac{1}{5}\)

(B) \(\frac{1}{3}\)

(C) \(\frac{1}{2}\)

(D) \(\frac{2}{3}\)

(E) 1

Since it is right angles triangle and sides AH || BG || CF,the two triangle are similar(△AHE ,△CFE ).

Ratio of Area of similar triangles=Area formula for 2 similar triangles =>

ΔAHE/ΔCFE=AE²/CE²=1/3

Some points to remember about trapezoid properties:

• Median - The median of a trapezoid is a line joining the midpoints of the two legs.

• The median line is always parallel to the bases.

• The median line is halfway between the bases.

• The median divides the trapezoid into two smaller trapezoids each with half the altitude of the original.

Since the median of a trapezoid is a line joining the midpoints of the two legs and B is the midpoint of the leg AC ,so BG is median,Also AH || BG || CF.

And since median divides the trapezoid into two smaller trapezoids each with half the altitude of the original.

So ratio of trapezoids=equal

Hence ratio of shaded to unshaded=1/2

_________________

Thanks,

Ankit

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