HOT Competition 1 Sep/8PM: A rectangle is divided into four regions wi

Question

https://gmatclub.com/forum/download/file.php?id=57261 A rectangle is divided into four regions with the areas of 1, 3, 4, and X, as shown above. What is the value of X? A. 1 B. 2 C. 3 D. 4 E. Cannot be determined. M37-103 2020-08-24_18-46-02.png

Tanushka77 · Accepted Answer

Two triangles (with areas 1 and 4) are similar because of angles. 
Ratio of their areas=4:1
Then Ratio of their sides=2:1

Then draw a height. 
(x*h)/2=1
Xh=2
The area of the rectangle then would be 3h*2x=6xh=12
Subtracting other regions we get 12-(1+3+4)=4

The answer is D

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piyush0122 · Answer

Let us take the triangles with area 1 and area 4. The two triangles will be similar since, one pair of angles are vertically opposite and the other angles are alternate interior angles, since the sides of rectangle are paralles.

Ratio of areas of the two triangles is 4:1.

Ratio of sides of the triangles is 2:1.

Let the side of the smaller triangle be y and height be h. so area =1/2 * y *h = 1
 => y*h=2

Now the side of the triangle with area 4 is 2y and height is 2h. So one side of the rectangle will be 2y and the other side will be h + 2h= 3h

Area of rectangle =3h * 2y =6hy = 6* (2) (since y*h =2)
 = 12
Area of rectangle is 12.
so area of x = 12 - ( 1 + 3 + 4)
 = 12 - 8
 =4
So, D is the correct answer.

Bunuel · Answer

Official Solution:

https://gmatclub.com/tests-beta/resources/import/img/M37-103.png 
 
A rectangle is divided into four regions with the areas of 1, 3, 4, and X, as shown above. What is the value of X? (Figure not drawn to scale.)

A.  1 
B.  2 
C.  3 
D.  4 
E. Cannot be determined

Consider yellow and green triangles below:
 
  https://gmatclub.com/tests-beta/resources/import/img/M37-103-2.png 
 
In these triangles, red angles are equal because they are a pair of opposite angles formed by intersecting lines and blue angles are equal because they are internal angles formed by a line crossing two parallel lines (opposite sides of a rectangle are parallel). So, all three angles in these triangles are equal, which makes these triangles 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 area of green triangle to that of red triangle is  4:1 , then the ratio of their corresponding sides must be  2:1 . So, if the base of the yellow triangle is  a , then the base of the green triangle is  2a  and if the height of the yellow triangle is  b , then the height of the green triangle is  2b .
 
We are given that the area of the yellow triangle is 1, so  \frac{1}{2}*ab=1  and thus,  ab=2 .
 
Next, notice that the length and width of the rectangle in this case would be  2a  and  b+2b=3b , and the area would be  (2a)*(3b)=6ab . Since  ab=2 , then the area of the rectangle is  6ab=12 .
 
Finally, the unknown area is  X=12-(4+3+1)=4 .

Answer: D

dep · Answer

IMO D

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mSKR · Answer

1. Sum of areas all : 8+x= ac
2. Sum of 1+3; Ab= 8
3. Area of 4: Cd= 8
4. Area of 1 : 1/2(a-d)e= 1
5. Sum of 1+x : 1/2a(c-b+e) = 1+x
6. Sum of 3&4 : a(b-e)+1/2 a (c-b+3)= 7
Find x?

5’: ac-ab+ae = 2(1+x)
From 1, 2
(8+x)- 8+ae = 2+2x--> ae= 2+x

6. Sum of 3&4 : a(b-e)+1/2 a (c-b+e)= 7
2ab-2ae+ac-ab+ae=14(from 2,1,5’)
ab-ae+ac=14-->no output

7. Sum of x&4 : a(c-b)+1/2 b(a) = 4+x
Ac-ab/2=4+x--> 8+x-4 no output

Value of x depends on 2 variables: e and c

Hence value of X can not be determined

JonShukhrat · Answer

The answer is D

Posted from my mobile device

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HOT Competition 1 Sep/8PM: A rectangle is divided into four regions wi

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