A chord PQ of a circle with radius r subtends an angle B at the center. Show that the area of the minor segment PQ=1/2.r^2.(B-sinB) and write down the area of the major segment PQ interms of r and B.
1/2 of the chord is r sin(B/2)
So, the area of the triangle is
2 * (1/2 r sin(B/2)(r cos(B/2))
= r^2 sin(B/2)cos(B/2)
= 1/2 r^2 sinB
The area of the whole sector is 1/2 r^2 B
So, the segment's area is as noted above: the sector minus the triangle.
To show that the area of the minor segment PQ is given by 1/2*r^2*(B - sinB), we can follow these steps:
Step 1: Draw the circle with center O and radius r.
Step 2: Draw the chord PQ, which subtends an angle B at the center O.
Step 3: Draw the radii OP and OQ, forming two sectors OPQ and OQP.
Step 4: The area of the minor segment PQ is equal to the difference between the area of the sector OPQ and the triangle OPQ:
Area of minor segment PQ = Area of sector OPQ - Area of triangle OPQ
Step 5: The area of a sector of a circle with radius r and angle B can be calculated as (B/360) * π * r^2. Therefore, the area of the sector OPQ is:
Area of sector OPQ = (B/360) * π * r^2
Step 6: The triangle OPQ is an isosceles triangle with two equal sides of length r. The base angle of this triangle is B, and the opposite side is r. We can use the formula for the area of a triangle:
Area of triangle OPQ = 1/2 * r * r * sin(B)
Step 7: Substituting the values from Step 5 and Step 6 into Step 4, we get:
Area of minor segment PQ = (B/360) * π * r^2 - 1/2 * r * r * sin(B)
Step 8: Simplify the equation:
Area of minor segment PQ = 1/2 * r^2 * [(B/180) * π - sin(B)]
Step 9: Since (B/180) * π = B * (π/180) = B * π/180, we can rewrite the equation as:
Area of minor segment PQ = 1/2 * r^2 * (B - sin(B * π/180))
Therefore, the area of the minor segment PQ is given by 1/2 * r^2 * (B - sin(B * π/180)).
For the area of the major segment PQ, we can follow a similar approach:
Step 1: The area of the major segment PQ is equal to the difference between the area of the circle and the area of the minor segment PQ.
Area of major segment PQ = Area of circle - Area of minor segment PQ
Step 2: The area of the circle can be calculated as π * r^2.
Step 3: Substituting the values into Step 1, we get:
Area of major segment PQ = π * r^2 - 1/2 * r^2 * (B - sin(B * π/180))
Step 4: Simplify the equation:
Area of major segment PQ = r^2 * (π - (B - sin(B * π/180))/2)
Therefore, the area of the major segment PQ is given by r^2 * (π - (B - sin(B * π/180))/2).