In a circle with radius 20cm, a chord is drawn with length 12 cm.
Find the area of the two regions created.
Find the perimeter of the two regions created.
I drew the 12 cm chord at the end of a radius, giving me an isosceles triangle with sides 20,20 and 12
Define the "two regions" created.
Hmmm. Isn't every chord drawn at the end of a radius?
Any chord divides the circle into two regions.
The area of the segment is the area of the sector less the area of the triangle, or r^2/2 (θ - sinθ)
sin θ/2 = 6/20, so θ/2 = .3047
sinθ = .5723
The area of the segment is thus 200*(.6094-.5723) = 7.42
So, the rest of the circle has area 400π - 7.42 = 1249.2171
arc length subtended by chord: rθ = 20*.6094 = 12.188
other arc is 2πr - rθ = 40π - 12.188 = 113.476
so, the perimeters are arc length + chord length = 24.188 and 125.476
(assuming no stupid arithmetic errors)
To find the area and perimeter of the two regions created by the chord in a circle with radius 20 cm, we need to find the area and perimeter of both the segment and the sector.
Segment:
1. First, we need to find the central angle of the segment. To do this, we can use the formula: θ = 2 * sin^(-1)(0.5 * (chord length / radius)).
In this case, the chord length is 12 cm and the radius is 20 cm.
θ = 2 * sin^(-1)(0.5 * (12 / 20)) = 2 * sin^(-1)(0.3) ≈ 2 * 17.46° ≈ 34.92°.
2. Next, we can find the area of the segment by using the formula: A = (θ/360) * π * r^2 - 0.5 * r^2 * sin(θ).
Substituting the values we know, A = (34.92/360) * π * 20^2 - 0.5 * 20^2 * sin(34.92°) ≈ 123.68 cm^2.
3. The perimeter of the segment is the length of the chord plus the arc length of the segment.
The arc length can be calculated as: arc length = (θ/360) * 2 * π * r.
Substituting the values, arc length = (34.92/360) * 2 * π * 20 ≈ 36.64 cm.
Therefore, the perimeter of the segment = chord length + arc length = 12 + 36.64 ≈ 48.64 cm.
Sector:
1. The central angle of the sector is the same as the central angle of the segment, θ ≈ 34.92°.
2. The area of the sector can be calculated using the formula: A = (θ/360) * π * r^2.
Substituting the values, A = (34.92/360) * π * 20^2 ≈ 71.96 cm^2.
3. The perimeter of the sector is equal to the length of the arc. Therefore, the perimeter of the sector = (θ/360) * 2 * π * r ≈ 36.64 cm.
So, the answers are:
Area of the segment ≈ 123.68 cm^2
Perimeter of the segment ≈ 48.64 cm
Area of the sector ≈ 71.96 cm^2
Perimeter of the sector ≈ 36.64 cm