Find the difference of the areas of two segments of a circle formed by a chord of length 5cm subtending an angle of 90degree at the centre.
a1 = 1/2 r^2 (θ-sinθ) = 25/2 (pi/2 - 1)
a2 = pi r^2 - a1 = 25pi - 25/2 (pi/2 - 1) = 25/4 (3pi + 2)
a2 - a1 = 25/2 (pi+2)
To find the difference of the areas of two segments of a circle formed by a chord, we need to follow these steps:
Step 1: Determine the radius of the circle.
In this case, we are given the length of the chord, which is 5 cm and the angle at the center, which is 90 degrees. We can use the formula:
radius = (chord length) / (2 * sin(angle/2))
radius = 5 / (2 * sin(90/2))
radius = 5 / (2 * sin(45))
radius = 5 / (2 * √2 / 2)
radius = 5 / √2
radius = 5√2 / 2√2
radius = 5√2 / 2
Step 2: Calculate the area of the major segment.
To calculate the area of the major segment, we need to find the area of the sector and subtract the area of the corresponding triangle. The central angle is 90 degrees, so the angle of the sector is also 90 degrees.
Area of the sector = (angle / 360 degrees) * π * (radius^2)
Area of the sector = (90 / 360) * π * ((5√2 / 2)^2)
Area of the sector = (1 / 4) * π * (25 / 2)
Area of the sector = π * 25 / 8
The triangle formed by the chord and two radii is a right-angled triangle. The angle at the center is 90 degrees, so the angle at the edge between one radius and the chord is 45 degrees. Since the chord is divided into two equal parts, the triangle is isosceles. Therefore, the two equal angles are 45 degrees.
Area of the triangle = (base * height) / 2
Area of the triangle = (5 * 5) / 2
Area of the triangle = 25 / 2
Area of the major segment = Area of the sector - Area of the triangle
Area of the major segment = π * 25 / 8 - 25 / 2
Step 3: Calculate the area of the minor segment.
To calculate the area of the minor segment, we need to subtract the area of the major segment from the area of the whole circle.
Area of the whole circle = π * radius^2
Area of the whole circle = π * (5√2 / 2)^2
Area of the whole circle = π * (25 / 2)
Area of the whole circle = 25π / 2
Area of the minor segment = Area of the whole circle - Area of the major segment
Area of the minor segment = 25π / 2 - (π * 25 / 8 - 25 / 2)
Step 4: Calculate the difference of the areas of the two segments.
Difference of the areas = Area of the major segment - Area of the minor segment
Now you can calculate the difference of the areas using the formulas and values derived above.