12 Days of Christmas GMAT Competition - Day 4: Right triangle ABC, rig

Question

12 Days of Christmas GMAT Competition with Lots of Fun

Right triangle ABC, right angled at B, is rotated clockwise, so that vertex B is at B', vertex A is at A' and vertex C is at C', as shown above. What is the ratio of the area of two green regions to the area of blue region?

A. 1:1
B. 2:1
C. 3:1
D. 4:1
E. 5:1

M37-76

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M37-76

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Official Answer

B

dep · Accepted Answer

Right triangle ABC, right angled at B, is rotated clockwise, so that vertex B is at B', vertex A is at A' and vertex C is at C', as shown above. What is the ratio of the area of two green regions to the area of blue region?

IMO B

Attachments

IMG_0333.jpg [ 587.4 KiB | Viewed 5525 times ]

IMO B

Bunuel · Answer

Official Solution:

https://gmatclub.com/tests-beta/resources/import/img/M37-76.png 
 
Right triangle ABC, right angled at B, is rotated clockwise, so that vertex B is at B', vertex A is at A' and vertex C is at C', as shown above. What is the ratio of the area of two green regions to the area of blue region?

A.  1:1 
B.  2:1 
C.  3:1 
D.  4:1 
E.  5:1

First of all, notice that (blue region) + (green region on the left) = (triangle ABC) and (blue region) + (green region on the right) = (triangle A'B'C'). Since triangles ABC and A'B'C' are congruent, then (green region on the left) = (green region on the right).
 
Now, consider triangle DBC below:
 
  https://gmatclub.com/tests-beta/resources/import/img/M37-76-2.png 
 
In that triangle  \angle ACB = \angle A'C'B'  (those yellow angles are the same angle just rotated), so  DB = DC .
 
Notice that (yellow angle at C) + (red angle at A) is 90° AND (yellow angle at B) + (red angle at B) is also 90°, so two red angles are equal. This means that  DA=DB  (D is the midpoint of AC). 
 
Drop perpendicular from D to BC. Triangles ABC and DFC are similar (because all their angles are equal) and DC is half of AC, thus DF is also half of AB. 
 
The bases of triangles ABC and DFC are the same (BC) and the height of ABCt (AB) is twice that of the height of DBC (DF), so the area of ABC is twice the area of DBC. Thus, the area of ADB (green region on the left) is equal to the area of DBC (blue region)
 
We know that the green regions are equal, so the ratio of the area of two green regions to the area of blue region is  2:1

Answer: B

Gmatfox · Answer

https://gmatclub.com/forum/download/file.php?id=59432 
Right triangle ABC, right angled at B, is rotated clockwise, so that vertex B is at B', vertex A is at A' and vertex C is at C', as shown above. What is the ratio of the area of two green regions to the area of blue region?

A. 1:1
B. 2:1
C. 3:1
D. 4:1
E. 5:1

Assume AC and B'C' intersect at point D. Check the picture attached.
We have: Angle B'C'A' = Angle BCA (Triangle ABC is similar to triangle A'B'C') --> Triangle BDC is an isosceles triangle with DB = DC
But we already know that ABC is a right triangle (angle B = 90 degree) --> D is the midpoint of AC
--> Area of blue region = Area of each green region = 1/2 area of whole triangle ABC = 1/2 area of whole triangle A'B'C'
--> Area of green regions : area of blue region = 2 : 1
Correct answer: B

vivek1408 · Answer

This question took me a good 15 mins to solve.. phew.
But I think the answer is B

To make our life easier we can assume the triangle ABC to be right isoceles, making two sides equal (lets say length X)and angles 45-45-90.
In this case, area of the left green triangle is 1/2*X*X/2 = X^2/4
and the area of the blue triangle is also 1/2*X*X/2 = X^2/4
since area(ABC) = area(A'B'C') =>therefore the area of the right green quadrilateral = area of left green triangle =X^2/4

Hence total green area = (X^2)/2
total blue area = (X^2)/4
making green:blue ratio = 2:1

answer B

vv65 · Answer

My answer is B 2:1

Here is my method:

Assume that the green triangle is a 90-60-30 triangle with hypotenuse=6 and small side=3
The rest of the working is in the diagram

Will appreciate a faster way of doing this question, as this method took too long. If I had drawn the diagram, it would have taken even longer

Eswar42 · Answer

IMO, answer is B, 2:1
it is derived from the angles that ABC is isosceles right angled triangle.
the triangles shaded in blue and green of ABC are equilateral triangles, see the attached sheet
the area of blue in ABC is equal to area of green in ABC. the green area in each orientation is same.
so answer is 2*green area of one / blue area = 2:1

QuantMadeEasy · Answer

As the triangle ABC is rotated about a point according to the above condition, the three regions will have equal areas
This can be understood by taking triangle ABC as 45-45-90 and related sides as x-x-root(2x)

ie. area of one green region = area of blue region
hence, ratio of area of total green region to area of blue region = 2:1

IMO B

DJ2911 · Answer

Answer: B (2:1)

Refer to the image attached below:
We have added a perpendicular line to BC from E meeting BC at F
Assume:
Angle A'C'B' = Angle ACB = theta 
BC = b
Note that since Area(EBC) is common to both the big triangles, Area(AEB) = Area(CEB'A') 
Area of two green regions = Area of AEB *2
Area of blue region = Area of EBC
Since EBC is an isosceles triangle: 
Area EBC = 1/2*b*b/2*Tan(theta) = (b^2 * tan(theta))/4
Area AEB = Area ABC - Area EBC = (b^2*tan(theta))/2 - (b^2tan(theta))/4 = (b^2*tan(theta))/4
Thus total area of green region / area of red region = 2* Area ABC / Area EBC = 2/1

rocky620 · Answer

Kindly Refer the Image Below:

Angle ACB = Angle A'C'B' = x,  So OB = OC (side opposite to equal angles)......eq1 
Angle BAC = y

Angle ACB + Angle BAC = x+y = 90 .......eq2 
Since, Angle ABO + Angle OBC = 90, so Angle ABO = y (from Eq2)

Since, Angle ABO = Angle OAB = y,  OB = OA.....eq3

From eq1 and eq3,  OA = OB = BC

We can deduce that OB is a median of AB, and a median divides the triangle in two equal parts with equal area.

Since triangle OBC is common for ABC and A'B'C', area of quad COB'A' = area of triangle ABO.

Now the two green parts and the blue part each have equal areas, so the ratio of area of green part to the blue part will be  2:1

IMO Option B

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