Find the area of the segment AYB shown in Fig. 11.6, if radius of the circle is 21 cm and ∠AOB=120∘. (Use π=722)
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Step-by-Step Solution
Step 1: Calculate the Area of the Sector AOB
To find the area of the segment AYB, we first need to calculate the area of the sector AOB. The formula for the area of a sector with angle θ and radius r is given by 360∘θ×πr2. Here, the angle θ=120∘ and the radius r=21 cm.
Step 2: Substitute values and compute sector area
Substitute the given values into the formula. We have θ=120∘, r=21 cm, and π=722. Let's perform the calculation:
Area of sector AOB=31×722×441=31×22×63=22×21=462 cm2.
Step 3: Calculate the Area of Triangle AOB
Next, we calculate the area of the triangle AOB. Since OA and OB are radii, △AOB is an isosceles triangle. The formula for the area of a triangle with two sides r and r and included angle θ is 21r2sinθ. Here, r=21 cm and θ=120∘.
Step 4: Substitute values and compute triangle area
Substitute the values: r=21 cm and θ=120∘. We know that sin(120∘)=sin(180∘−60∘)=sin(60∘)=23.
Area of △AOB=21×441×23=44413 cm2.
Step 5: Calculate the Area of the Segment AYB
The area of the minor segment AYB is obtained by subtracting the area of the triangle AOB from the area of the sector AOB. We have already calculated both these areas in the previous steps.
Step 6: Substitute values and compute segment area
Substitute the calculated areas:
Area of segment AYB=462−44413 cm2. This is the exact area of the segment AYB.