Question:In the figure above, points A, B, C, D, and E lie on a line. A is on both circles, B is the center of the smaller circle, C is the center of the larger circle, D is on the smaller circle, and E is on the larger circle. What is the area of the region inside the larger circle and outside the smaller circle?
(1) AB = 3 and BC = 2
(2) CD = 1 and DE = 4
A question like this may seem to feel like easy and on reading it usually students end up marking option A as answer! Simply because the right side of our brain could easily decode statement 1 but NOT statement 2.
A GMAT Geometry question like this needs a
very wide but precise perspective and inference making skills.If you haven’t got it right and hence scrolling through the answers lets break this down for you in 3 simple steps or GMAT Track of thoughts.
GMAT Track of Thought 1What do I have from the Question stem? Do I have the center mentioned in the question? Yes I do! B is the CENTER of the smaller circle, C is the CENTER of the larger circle.
Points A,D,E are just points on the circumferences of the larger and smaller circles respectively.
What do I need to answer?The area of the region inside the larger circle and outside the smaller circle. Let me shade it.
GMAT Track of Thought 2To answer the question stem what variables or constants I need?
The shaded area is the difference between the areas of the larger circle with centre C and the smaller circle with centre B. Let me mark the radiuses of the larger and smaller circle as R and r respectively.
So I would need to know
∏\(R^2\) - ∏\(r^2\) and since ∏ is a constant I need the difference between \(R^2 \)and \(r^2 \)or the values of R and r individually so that the difference could be found.
GMAT Track of Thought 3What do I have from statement 1? AB=3 & BC =2
AB=3 provides me with the value of “r”.
I need R which is dependent on centre C. Since BC=2, I can add AB and BC to have AC.
So AC=3+2=5
I would do so since AC is the radius of the larger circle and gives me exactly the value of‘R’!
So I have r = 3 and R = 5.
This is sufficient.
I can
eliminate B,C,E answer choices right away.
GMAT Track of Thought 4What do I have from statement 2?CD = 1 and DE = 4
Its observed that CD and DE added gives me CE which is the radius of the larger circle.
So I have CD + DE = 1 + 4 = 5= CE as R.
Now how do I think for ‘r’?
Here is where my observation and logical skills need to work in tandem. Let me see. I have to be careful to not intermix data from statement 1 here. May be redrawing the figures is a good idea to avoid this error!Since CE = 5 AC also is 5 (R).Diameter of the larger circle = 2 * 5 = 10.Doesnt help!
Since CD =1 , AD = 5 +1 =6 Bam! Doesn’t help yet!
Lets think harder.
Well AD is the diameter of the smaller circle(visual skills)
This implies AD = 2*r
6 = 2*r or r = 3 Bingo!
So I have the values of R & r from statement 2 also and hence it is sufficient to answer the question stem.
Eliminate A and mark answer option D as the correct answer to this question.
PS-If this question was a miss for you try to absorb the takeaway and practice more questions(official) that challenges your visual and logical skills. May be pick up some Advanced 700 questions from Geometry after you have mastered the Intermediate.Devmitra Sen
GMAT Mentor _________________