Find the perimeter and area of a segment of a circle of radius 8cm, cut off by a chord that subtended a central angle of:. a 120o. b 3/4

Answer 1

a) To find the perimeter and area of a segment of a circle, we need to know the length of the chord and the height of the segment.

We are given that the radius is 8cm.

Using the central angle of 120o, we can divide the circle into two equal parts, each containing an angle of 60o. The chord that subtends this angle will bisect the circle into two equal parts. This chord is also the diameter of the circle, and its length can be found using the formula:

diameter = 2 x radius = 2 x 8cm = 16cm

To find the length of the chord that subtends the central angle of 120o, we can use the following formula:

length of chord = 2 x radius x sin(angle/2)

where angle is the central angle in degrees.

Substituting in the values, we get:

length of chord = 2 x 8cm x sin(60o) ≈ 13.86cm

Now, we need to find the height of the segment, which is the distance between the chord and the center of the circle.

We can divide the segment into two right triangles, with the chord as their hypotenuse. The height is the perpendicular drawn from the center of the circle to the chord, bisecting it.

Using one of the triangles, we can find the height as:

height = radius x cos(angle/2)

Substituting in the values, we get:

height = 8cm x cos(60o/2) = 8cm x cos(30o) ≈ 6.93cm

Now, we can calculate the perimeter and area of the segment using the following formulas:

perimeter = length of chord + 2 x height
area = (angle/360) x π x radius2 - 0.5 x length of chord x height

Substituting in the values, we get:

perimeter = 13.86cm + 2 x 6.93cm ≈ 27.72cm
area = (120/360) x π x 82 - 0.5 x 13.86cm x 6.93cm ≈ 88.35cm2

Therefore, the perimeter of the segment is approximately 27.72cm and its area is approximately 88.35cm2.

b) We are given that the central angle is 3/4 of the total angle, which means it is 270o.

Using the same method as above, we can find the length of the chord as:

length of chord = 2 x 8cm x sin(270/2) ≈ 15.45cm

The height of the segment can be found as:

height = 8cm x cos(270/2) = 8cm x cos(135o) ≈ 2.83cm

Using the formulas for perimeter and area of a segment, we get:

perimeter = 15.45cm + 2 x 2.83cm ≈ 21.11cm
area = (270/360) x π x 82 - 0.5 x 15.45cm x 2.83cm ≈ 39.27cm2

Therefore, the perimeter of the segment is approximately 21.11cm and its area is approximately 39.27cm2.