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

Official Solution:




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:



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

IMO D

Posted from my mobile device

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

The answer is D

Posted from my mobile device

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