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