A chord PQ of length 5.6cm divides a circle of radius 3.5cm into two segments.calculate the difference in areas between major and minor segment in the circle of chord PQ.
Draw a diagram. If the angle subtending the chord is x,
sin(x/2) = 2.8/3.5 = 0.8, so x/2 = 0.927
The area of a segment is 1/2 r^2 (x-sinx)
So, the smaller segment has area 1/2 * 3.5^2 * (2*.927-.8) = 6.46
The area of the whole circle is pi * 3.5^2 = 38.48
so, the area of the larger segment is 32.02
To find the difference in areas between the major and minor segments of a circle with chord PQ, we need to calculate the areas of both segments.
1. The first step is to find the length of the perpendicular bisector of the chord PQ. This bisector, together with the chord PQ, divides the circle into two equal parts, with each part being a semicircle.
The perpendicular bisector of a chord passes through the center of the circle. Since O is the center of the circle, the perpendicular bisector passes through O.
Therefore, the length of the perpendicular bisector, which is the diameter of the circle, is equal to twice the radius (2 × 3.5 cm = 7 cm).
2. Now that we know the diameter, we can calculate the areas of the major and minor segments.
The major segment is a sector of the circle minus the triangle formed by the chord.
The angle subtended by the major segment at the center of the circle is twice the angle subtended by the minor segment.
To find the angle subtended by the major segment (θ), we can use the formula:
θ = 2 × arcsin(length of chord / diameter)
θ = 2 × arcsin(5.6 cm / 7 cm)
Now, we can calculate the area of the major segment using the formula:
Area of major segment = (θ/360°) × πr^2 - (1/2) × chord-length × (diameter - chord-length)
3. To find the area of the minor segment, we can subtract the area of the major segment from the area of the original circle.
Area of the minor segment = Area of the circle - Area of the major segment.
By calculating these values, we can determine the difference in areas between the major and minor segments of the circle with chord PQ.
To find the difference in areas between the major and minor segments of a circle, you'll need to know the length of the chord and the radius of the circle. In this case, the chord length is given as 5.6cm, and the radius is given as 3.5cm.
First, let's find the distance between the chord and the center of the circle. This distance is also known as the perpendicular distance or the height of the segment. To find this distance, we can use the Pythagorean theorem.
The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is the radius (3.5cm), and the other two sides are half of the chord length (5.6cm/2 = 2.8cm) and the perpendicular distance we're trying to find (let's call it h).
So we have:
(2.8cm)^2 + h^2 = (3.5cm)^2
Simplifying this equation gives us:
7.84cm^2 + h^2 = 12.25cm^2
Subtracting 7.84cm^2 from both sides, we get:
h^2 = 12.25cm^2 - 7.84cm^2
h^2 = 4.41cm^2
Taking the square root of both sides, we find:
h = 2.1cm
Now that we know the height of the segment, we can calculate the areas of the major and minor segments. The formula for the area of a segment of a circle is given by the difference in areas between the corresponding sector and triangle.
The area of the major segment is the area of the sector minus the area of the triangle. The formula for the area of a sector is (θ/360) * π * r^2, where θ is the central angle. In this case, the central angle can be found using the sine function: θ = 2 * arcsin(h/r).
The area of the minor segment is the area of the circle minus the area of the major segment.
Let's calculate these areas:
1. Area of the major segment:
- Central angle θ = 2 * arcsin(h/r)
= 2 * arcsin(2.1cm/3.5cm)
≈ 2 * arcsin(0.6)
≈ 2 * 36.87°
≈ 73.74°
- Area of the sector = (θ/360) * π * r^2
= (73.74°/360°) * π * (3.5cm)^2
≈ (0.205) * 3.14 * 12.25cm^2
≈ 7.97cm^2
- Area of the triangle = (1/2) * base * height
= (1/2) * 5.6cm * 2.1cm
≈ 5.88cm^2
- Area of the major segment = Area of the sector - Area of the triangle
= 7.97cm^2 - 5.88cm^2
≈ 2.09cm^2
2. Area of the minor segment:
- Area of the circle = π * r^2
= 3.14 * (3.5cm)^2
≈ 38.465cm^2
- Area of the minor segment = Area of the circle - Area of the major segment
= 38.465cm^2 - 2.09cm^2
≈ 36.375cm^2
Now, the difference in areas between the major and minor segments is:
Difference = Area of the minor segment - Area of the major segment
= 36.375cm^2 - 2.09cm^2
≈ 34.285cm^2
Therefore, the difference in areas between the major and minor segments in the circle of chord PQ is approximately 34.285 square centimeters.