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

Question

https://gmatclub.com/forum/download/file.php?id=42350 If riangle{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 New question self made TRIANGLE.png

Princ · Accepted Answer

OA: C
 	riangle{HAE} , 	riangle{GBE}  and  	riangle{FCE}  are similar as  AH || BG || CF  and  \angle E  is common in three  	riangle .
AB=BC=CE=√3 , BE = CE+BC = 2√3 , AE = AB+BC+CE = 3√3.
 \frac{{Area of 	riangle {FCE}}}{{Area of 	riangle {HAE}}}=\frac{{CE^2}}{{AE^2}}= \frac{{(\sqrt {3})^2}}{{(3\sqrt {3})^2}}=\frac{1}{9} 
 \frac{{Area of 	riangle {GBE}}}{{Area of 	riangle {HAE}}}=\frac{{BE^2}}{{AE^2}}= \frac{{(2\sqrt {3})^2}}{{(3\sqrt {3})^2}}=\frac{4}{9} 
Area of colored portion  GBCF  = Area of  	riangle {GBE}  -Area of  	riangle {FCE}  =  (\frac{4}{9}-\frac{1}{9}) Area of  	riangle {HAE} = \frac{1}{3} Area of  	riangle {HAE} 
Area of Uncolored Portion = Area of  	riangle {HAE}  - Area of colored Portion =  \frac{2}{3} Area of  	riangle {HAE} 
 \frac{{Area of Colored portion}}{{Area of Uncolored Portion}}=\frac{\frac{1}{3}Area of 	riangle {HAE}}{\frac{2}{3}Area of 	riangle {HAE}} =\frac{1}{2}

chetan2u · Answer

kudos to best solutions..
reserved for OE

Nikhil · Answer

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

chetan2u · Answer

sorry for the typo
edited

Nikhil · Answer

Area of trapezium BGCF =  (BG + CF) * \frac{\sqrt{{3}}}{2}

Area of trapezium AHBG =  (AH + BG) * \frac{\sqrt{{3}}}{2}

Area of triangle CFE =  CF * \frac{\sqrt{{3}}}{2}

Area of trapezium AHBG + Area of triangle CFE =  (AH + BG) * \frac{\sqrt{{3}}}{2}  +  CF * \frac{\sqrt{{3}}}{2}

Ratio of area of colored portion to non-colored portion =  \frac{(BG + CF) * \frac{\sqrt{{3}}}{2}}{{(AH + BG) * \frac{\sqrt{{3}}}{2} + CF * \frac{\sqrt{{3}}}{2}

Ratio of area of colored portion to non-colored portion =  \frac{BG + CF}{BG + CF + AH}

Triangles AHE and CFE are similar since  \angle HAE = \angle FCE = 90  and  \angle AEH = \angle CEF

=> The sides of these triangles are in proportion =>  \frac{AH}{CF} = \frac{AE}{CE} = 3

=> AH = 3 CF

Triangles BGE and CFE are similar since  \angle GBE = \angle FCE = 90  and  \angle BEG = \angle CEF

=> The sides of these triangles are in proportion =>  \frac{BG}{CF} = \frac{BE}{CE} = 2

=> BG = 2 CF

Substituting AH and BG values in the ratio, we get

Ratio of area of colored portion to non-colored portion =  \frac{3CF}{6CF}  =  \frac{1}{2}

Hence option C

StudiosTom · Answer

https://gmatclub.com/forum/download/file.php?id=42350 
If  	riangle{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

pandeyashwin · Answer

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

nati1516 · Answer

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

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

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

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