A chord of length 10 cm divides a circle of radius 5root2 cm in two segments. find the area of minor segment.
You know that the area of a segment is
a = 1/2 r^2 (θ-sinθ)
Draw a diagram. It is clear that
sin θ/2 = 5/5√2, so θ/2 = π/4
Now plug and chug
To find the area of the minor segment, we first need to find the angle subtended by the chord at the center of the circle.
1. Draw the radius from the center of the circle to one of the endpoints of the chord.
2. Since the radius is perpendicular to the chord, the triangle formed by the radius, the chord, and the diameter of the circle is a right triangle.
3. The chord is divided into two equal parts by the radius, so each part is 5 cm.
4. The hypotenuse of the right triangle is the radius of the circle, which we are given as 5√2 cm.
5. Apply the Pythagorean theorem: (5√2)^2 = 5^2 + 5^2
Simplifying, we get: 50 = 25 + 25
Therefore, the hypotenuse is indeed the radius of the circle.
6. The right angle is formed by the radius and one half of the chord, so the other angle is the angle subtended by the chord at the center of the circle.
Once we find the angle, we can calculate the area of the minor segment.
7. The angle θ can be found using the sine of the angle: sin(θ) = Opposite/Hypotenuse = 5/5√2 = 1/√2
8. Taking the inverse sine of both sides: θ = sin^(-1)(1/√2) ≈ 45°
Note: When taking the inverse sine, make sure the calculator is set to degree mode.
9. The area of the minor segment is the area of the sector minus the area of the triangle formed by the radius, the chord, and the diameter.
The area of the sector is (θ/360°) × πr^2, and the area of the triangle is (1/2) × base × height.
10. The radius (r) is 5√2 cm.
11. The base of the triangle is the length of the chord, which is 10 cm.
12. The height of the triangle can be found by bisecting the chord, creating a right triangle with one leg equal to half the length of the chord (5 cm) and the hypotenuse equal to the radius (5√2 cm).
We can use the Pythagorean theorem to find the height: height^2 = hypotenuse^2 - leg^2 = (5√2)^2 - 5^2 = 50 - 25 = 25
Taking the square root of both sides: height = √25 = 5 cm.
13. The area of the sector is (45°/360°) × π × (5√2)^2 = (1/8) × 50π = 25π/4
14. The area of the triangle is (1/2) × base × height = (1/2) × 10 × 5 = 25
15. Therefore, the area of the minor segment is (25π/4) - 25.