This question tests concepts of finding areas of semi circles and right isosceles triangles. The crux of solving this GMAT problem solving practice question is one's ability to see the shaded region and the remaining parts as geometrical shapes for which areas can be computed. Presented as part of Q-51 series by 4GMAT. Tough GMAT Quant practice questions.

1. Triangle and semicircle
Geometry
Problem Solving – Question 8

2. Question
In the figure given below, ABC and CDE are two identical semi-circles of radius 2 units. B
and D are the mid points of the arc ABC and CDE respectively. What is the area of the
shaded region?
A. 4π - 1
B. 3π - 1
C. 2π - 4
D.
3𝜋−1
2
E. 2π - 2

3. ◴Step 1 : Segregate the semicircle into shaded
region and the balance

4. ABC and CDE are two identical semi-circles of radius 2 units.
What is the area of the shaded region?
01 What does the semicircle comprise apart from the shaded region?
A C
B
E
D

5. ABC and CDE are two identical semi-circles of radius 2 units.
What is the area of the shaded region?
01 What does the semicircle comprise apart from the shaded region?
· Mark centers of the semicircles ABC and CDE
· Let the centers be P and Q.
A C
B
E
D
P Q

6. ABC and CDE are two identical semi-circles of radius 2 units.
What is the area of the shaded region?
01 What does the semicircle comprise apart from the shaded region?
· Mark centers of the semicircles ABC and CDE
· Let the centers be P and Q.
· In semicircle ABC, draw PB perpendicular to AC.
A C
B
E
D
P Q

7. ABC and CDE are two identical semi-circles of radius 2 units.
What is the area of the shaded region?
01 What does the semicircle comprise apart from the shaded region?
· Mark centers of the semicircles ABC and CDE
· Let the centers be P and Q.
· In semicircle ABC, draw PB perpendicular to AC.
· Half the semicircle, PBC comprises the shaded
region and a right triangle PBC.

8. ABC and CDE are two identical semi-circles of radius 2 units.
What is the area of the shaded region?
01 What does the semicircle comprise apart from the shaded region?
· Mark centers of the semicircles ABC and CDE
Radius of semi circle = 2 units.
Base and height of triangle = 2 units each.
· Let the centers be P and Q.
· In semicircle ABC, draw PB perpendicular to AC.
· Half the semicircle, PBC comprises the shaded
region and a right triangle PBC.
· PB and PC are radii to the semi circle.
So, the base and height of triangle PBC are the radii
of the semicircle.

9. ◴Step 2: Compute areas of half the semicircle
and the right triangle and find the answer

10. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
02 Compute areas of half of semicircle, PBC and area of right triangle PBC
·
Area of half of semicircle ABC, i.e., region PBC
=
1
2
πr2
2
=
π×22
4
=  sq. units

11. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
02 Compute areas of half of semicircle, PBC and area of right triangle PBC
·
Area of half of semicircle ABC, i.e., region PBC
=
1
2
πr2
2
=
π×22
4
=  sq. units
· Area of triangle PBC =
1
2
bh =
1
2
×2×2 = 2 sq. units

12. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
02 Compute areas of half of semicircle, PBC and area of right triangle PBC
·
Area of half of semicircle ABC, i.e., region PBC
=
1
2
πr2
2
=
π×22
4
=  sq. units
· Area of triangle PBC =
1
2
bh =
1
2
×2×2 = 2 sq. units
· Area of shaded region in semicircle ABC = ( - 2) sq. units

13. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
02 Compute areas of half of semicircle, PBC and area of right triangle PBC
·
Area of half of semicircle ABC, i.e., region PBC
=
1
2
πr2
2
=
π×22
4
=  sq. units
· Area of triangle PBC =
1
2
bh =
1
2
×2×2 = 2 sq. units
· Area of shaded region in semicircle ABC = ( - 2) sq. units
· Area of total shaded region = 2( - 2)
= (2 - 4) sq. units
Choice C is the
answer

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