What is the area of the minor segment cut off a circle of radius 10 cm by a chord of length 12 cm?

Could you please show me the working out for this question?

The answer in the textbook is 16 sq cm.

Thanks!!!

Make a sketch showing the chord of 12 and the the two radii of 10.

I see an isosceles triangle. Draw an altitude from the centre to that chord, making two congruent right-angled triangles.
let the height be x, then x^2 + 6^2 = 10^2
x^2 = 100-36 = 64
x = √64 = 8
So the area of the large triangle, 10,10,12 is
(1/2) (12)(8) = 48 cm^2

We have to find the central angle of the sector.
Let each angle at the centre of the right-angled triangles be Ø
sinØ = 6/10 = .6
Ø = 36.87‡
and the central angle is 2Ø = 73.74°

area of whole circle = π(10)^2 = 100π
area of sector/100π = 73.74/360
area of sector = 64.35 cm^2

sooo, the segment is 64.35 - 48 = 16.35 cm^2

(I carried all decimals my calculator could hold and only rounded off the final answer.)

To find the area of the minor segment cut off by a chord, we can follow these steps:

1. Calculate the central angle of the sector formed by the chord.
- Since the chord of length 12 cm cuts off a minor segment, it divides the circle into two sectors. The central angle can be found using the formula:
Central Angle (in radians) = 2 * arcsin(chord length / (2 * radius))

In this case:
Central Angle = 2 * arcsin(12 cm / (2 * 10 cm))

2. Calculate the area of the sector using the central angle.
- The formula to find the area of a sector is:
Area of Sector = (Central Angle / 2π) * π * (radius^2)

In this case:
Area of Sector = (Central Angle / 2π) * π * (10 cm)^2

3. Calculate the area of the triangle formed by the chord.
- The length of the chord (12 cm) is equal to the base of the triangle. To find the height of the triangle, we can use the formula:
Height = radius - (sqrt(radius^2 - (chord length / 2)^2))

In this case:
Height = 10 cm - (sqrt(10 cm^2 - (12 cm / 2)^2))

4. Calculate the area of the triangle using the base and height.
- The area of a triangle is given by the formula:
Area of Triangle = (base * height) / 2

5. Calculate the area of the minor segment.
- The area of the minor segment can be found by subtracting the area of the triangle from the area of the sector.
Area of Minor Segment = Area of Sector - Area of Triangle

By following these steps, we can find the area of the minor segment of the given circle.

To find the area of the minor segment, we need to first find the area of the sector and then subtract the area of the triangle formed by the chord.

Step 1: Find the area of the sector.
The area of a sector can be calculated using the formula:

Area of sector = (θ/360) * π * r^2

Where θ is the angle subtended by the sector at the center of the circle (in degrees), and r is the radius of the circle.

In this case, the angle θ can be found by using the chord length. Since the chord divides the circle into two equal parts, each part subtends an angle of 2θ at the center of the circle. So, we can find θ by using the formula:

θ = 2 * sin^(-1) (c / 2r)

Where c is the length of the chord (12 cm) and r is the radius of the circle (10 cm).

Substituting the values, we get:

θ = 2 * sin^(-1)(12 / (2 * 10))
= 2 * sin^(-1)(0.6)
≈ 2 * 36.87°
≈ 73.74°

Now, we can find the area of the sector:

Area of sector = (θ/360) * π * r^2
= (73.74°/360°) * π * (10 cm)^2
≈ (0.2048) * π * 100 cm^2
≈ 64.63 cm^2

Step 2: Find the area of the triangle.
The area of a triangle can be calculated using the formula:

Area of triangle = 0.5 * base * height

In this case, the base of the triangle is the chord length (12 cm). To find the height, we can use the Pythagorean theorem. The height is the perpendicular distance from the center of the circle to the midpoint of the chord. So:

height = √(r^2 - (c/2)^2)
= √(10^2 - (12/2)^2)
= √(100 - 36)
= √64
= 8 cm

Now, we can find the area of the triangle:

Area of triangle = 0.5 * base * height
= 0.5 * 12 cm * 8 cm
= 48 cm^2

Step 3: Find the area of the minor segment.
To find the area of the minor segment, subtract the area of the triangle from the area of the sector:

Area of minor segment = Area of sector - Area of triangle
= 64.63 cm^2 - 48 cm^2
= 16.63 cm^2

Therefore, the area of the minor segment is approximately 16.63 cm^2, which is very close to the answer given in the textbook (16 cm^2).