HOT Competition 3 Sep/8AM: In a triangle ABC, shown above, three lines

Question

https://gmatclub.com/forum/download/file.php?id=57263 Figure NOT drawn to scale. In a triangle ABC, shown above, three lines go across point O, such that the lines are parallel to the sides of triangle ABC. If the areas of the yellow, green and blue regions shown are 1, 4 and 9 respectively, what it the area of triangle ABC? A. 24 B. 28 C. 32 D. 36 E. 54 Kudos are rewarded only for explanations published within 1 hour of posting M37-104 HOT Geometry.png

Bunuel · Accepted Answer

Official Solution:

https://gmatclub.com/tests-beta/resources/import/img/M37-104.png 
 
In a triangle ABC, shown above, three lines go across point O, such that the lines are parallel to the sides of triangle ABC. If the areas of the yellow, green and blue regions shown are 1, 4 and 9 respectively, what it the area of triangle ABC? (Figure NOT drawn to scale.)

A.  24 
B.  28 
C.  32 
D.  36 
E.  54

All three sides in all three triangles are parallel to each other, hence these triangles are similar to each other.
 
 In similar triangles, if the sides are in the ratio  \frac{m}{n} , the areas of the triangles are in the ratio  (\frac{m}{n})^2 .  Since the ratio of the areas of the triangles is  1:4:9 , then the ratio of their corresponding sides must be  1:2:3 . So, if the base of the yellow triangle is  x , then the base of the green triangle is  2x  and the base of the blue triangle is  3x .
 
  https://gmatclub.com/tests-beta/resources/import/img/M37-104-1.png 
 
Next, the two grey figures in the image above are parallelograms, so their opposite sides are equal, which makes the base of ABC equal to  x+3x+2x=6x . 
 
The blue triangle is also similar to ABC, and since the ratio of their bases is  6x:3x=2:1 , then the ratio of their areas must be  4:1 . 
 
Finally, since the area of blue triangle is 9, then the area of ABC is  4*9=36 .

Answer: D

SohGMAT2020 · Answer

In a triangle ABC, shown above, three lines go across point O, such that the lines are parallel to the sides of triangle ABC. If the areas of the yellow, green and blue regions shown are 1, 4 and 9 respectively, what it the area of triangle ABC? (Figure NOT drawn to scale.)

A. 24
B. 28
C. 32
D. 36
E. 54

Solution:

Since all the three triangles are similar

The ratio of sides is in the ratio of Sqrt (1) : Sqrt (4) : Sqrt (9) = 1:2:3

So, if the base of the yellow triangle is a; base of green triangle = 2a ; base of blue triangle = 3a

and, if the height of the yellow triangle is b; heightof green triangle = 2b ; height of blue triangle = 3b

So, the total base of the main triangle = 6a; and the total height of the main triangle = 6b;

Area = 1/2 * 6a * 6 b
 = 18 ab

Given 1/2 a*b = 1 = ab = 2; therefore 18 ab = 36

IMO D is the right answer

Superman249 · Answer

IMO Ans Is D i.e. 36

All triangles i..e.yellow, green and blue are similar as line passing through O is parallel to bas of triangle all angle are equal

We know that 
Square of ratio of side = Area of Triangle

So let a , b & c be side of yellow, green & blue so 
a^2 : b^2 : c^2 = 1:4:9
a:b:c = 1:2:3
a:b:c = 1x:2x:3x

Base of triangle ABC
Will be a+b+c = 6x

Now applying 
Square of ratio of side = Area of Triangle
In comparison to blue triangle whose side is 3x

= (6x/3x)^2 = (Area required/ 9)
 Area required is 36

Posted from my mobile device

eldorado12 · Answer

Posted from my mobile device

Rogermoris · Answer

Whenever we encounter such questions that involve multiple triangles then it is a clear hint to use similarities of triangles to solve them. Refer to the diagram I have attached below and follow the steps as follows:

Because the lines through O are parallel to the sides of ABC, each of the three small triangles HIO, DEO, FGO is similar to ABC and so they are all similar to each other.
 A triangle’s area is proportional to its side length squared, so the square root of its area is proportional to its side length . We are given the information:

areas HIO:DEO:FGO= 9:1:4

Because the triangles are similar, their sides will have a ratio equal to the square root of their areas.

sides HIO:DEO:FGO = 3:1:2

Suppose HI = 3x. Because the triangles are similar, DO = x and OG = 2x.

Now notice BDOI is a parallelogram. This is because lines through O are parallel to the sides of ABC, so IO and BD are parallel and so are DO and BI. Opposite sides in a parallelogram have equal length, so BI = DO = x.

Similarly HOGC is a parallelogram, and HC = OG = 2x.

Consequently we can calculate BC has a length:

= BI + IH + HC
= 6x

Triangles ABC and HIO are similar, and their sides are in a ratio:

BC:IH
= (6x):(3x)
= 2:1

Consequently their areas will be in a ratio of the square of their sides:

area BC:IH
=  2^2 : 1^2 
= 4:1

Thus triangle ABC has an area 4 times as large as HIO, so its area is = 4*9 = 36

MrBaby · Answer

Let me show the explanation via attachment. Thanks.

Posted from my mobile device

Louis14 · Answer

IanStewart   VeritasKarishma 
Could you please explain how the three triangles and the larger triangle are similar? I do have a slight idea but I'm just not being able to see the clear picture at all. Please help. Thanks so much.

KarishmaB · Answer

Because DE || AC, and HJ is a transversal on them, corresponding angles x and y are equal. Because HJ and BC are parallel and DE is a transversal on them, corresponding angles y and z are equal. Because DE || AC, and BC is a transversal on them, corresponding angles z and c are equal. The same logic applies to the other set of angles (shown by ")" but not named to avoid confusion) By AA, the 4 triangles are similar. HOT%20Geometry.png

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HOT Competition 3 Sep/8AM: In a triangle ABC, shown above, three lines

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HOT Competition 3 Sep/8AM: In a triangle ABC, shown above, three lines

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