ABCDEFGH is a regular octagon inscribed in circle is r=1 unit O is the centre of circle. find 1.radian measure of angle OAB.2.length of chord AB.3.length of arc AB.4.Area of region enclosed between chord AB and arc AB.
Answers
Answer:
∠OAB = 3π/8 = 67.5°
AB = √(2 - √2) = 0.765
π/4 = 0.786
( π - 2√2 )/8 = 0.04 sq units
Step-by-step explanation:
Regular octagen has all its side equal
if we join center O with all vertices A , B , C ............H
we will get OA = OB = Radius = 1
We will get 8 segment
angle of each segment = 360°/8 = 45°
in Δ OAB
OA = OB => ∠OAB = ∠OBA
∠AOB = 45° = π/4
=> ∠OAB + ∠OBA + ∠AOB = π
=> 2 ∠OAB + π/4 = π
=> 2∠OAB = 3π/4
=> ∠OAB = 3π/8 = 67.5°
AB = √ (OA² + OB² - 2 OA * OB Cos ∠AOB)
=> AB = √ (1² + 1² - 2 Cos 45°)
=> AB = √(2 - √2)
=> AB = 0.765
.length of arc AB = (45/360) * (2 * π * 1)
= π/4
Area of region enclosed between chord AB and arc AB
= ( Arc area - Δ OAB Area)
Arc area = (45/360) π * 1² = π/8 sq unit
Δ OAB Area = (1/2) OA * OB * Sin∠AOB = (1/2) * 1 * 1/√2 = 1/2√2 sq unit
π/8 - 1/2√2
= ( π - 2√2 )/8 sq units