Triangle and semicircleGeometryProblem Solving – Question 8

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

◴Step 1 : Segregate the semicircle into shaded

region and the balance

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?

AC

B

E

D

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.

AC

B

E

D

P Q

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.

AC

B

E

D

P Q

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.

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.

◴Step 2: Compute areas of half the semicircle

and the right triangle and find the answer

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

= 12πr2

2= π×22

4= sq. units

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

= 12πr2

2= π×22

4= sq. units

· Area of triangle PBC = 12

bh = 12×2×2 = 2 sq. units

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

= 12πr2

2= π×22

4= sq. units

· Area of triangle PBC = 12

bh = 12×2×2 = 2 sq. units

· Area of shaded region in semicircle ABC = ( - 2) sq. units

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

= 12πr2

2= π×22

4= sq. units

· Area of triangle PBC = 12

bh = 12×2×2 = 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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