If triangle{AHE} is a right angle triangle and AH || BG || CF

Can be solved with ratio property. Area of triangle "ABC"/Area of triangle "DEF" = (Side of ABC)^2/(Side of DEF)^2

The problem statement and the diagram are little inconsistent. Do you mean to say AB = BC = CE = \(\sqrt{3}\) ?

For similar triangles: if corresponding sides increase by a scale factor of k, the area increases by a scale factor of k^2.

ΔHAE, ΔGBE, and ΔFCE are all similar triangles because AH || BG || CF.

Say area of ΔFCE is x.

Scale factor of BE/CE is 2, so area of these corresponding triangles must increase by a factor of 4. If area(ΔFCE)=x, than area(ΔGBE)=4x. Therefore, the area of quadrilateral GBCF is 3x.

Scale factor of AE/CE is 3, so area must increase by a factor of 9. If area(ΔFCE)=x, than area(ΔHAE)=9x. Therefore, the area of quadrilateral HACF is 8x. If quadrilateral GBCF is 3x, then the area of quadrilateral HABG is 5x.

area (shaded region/unshaded region) = 3x/(x+5x) = 3x/6x = 1/2

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sorry for the typo
edited

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

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