Answer:
A) This can be shown using the equation of Arc length. Arc length = r x Theta where r is the radius of the circle in this case OA and Theta is the angle subtended by the arc, which is angle AOC. This angle can be found because we know the angle OAC which is π/3 rad and angle OCA is π/2 (right angle) so that makes angle AOC π - (π/2+π/3) because the total angles should add up to π, we will find that the angle AOC is π/6. So now we know the arc length so we can substitute in the equation the Theta and the arc length to get 5 = r x π/6
dividing both sides by π/6 we will get 5/(π/6) which is approximately 9.55 cm.
B) To find the perimeter we need to add up all sides of the shaded region, in this case the arc length that we have (5) and add it with AC and BC. So the goal is to find AC and BC. We found OA to be 9.55 cm. We have angles so we can use trigonometric functions to find the other sides. we have the hypotneus so we can use either sin or cos so sin will give us length of AC so sin(π/6) = AC/9.55 and you will find that AC is 4.75 and cos(π/6) = OC/9.55 and you will find OC is 8.27
so now we know the perimeter is 5 + 4.75 + BC well we still need to find BC well we know that the radius of the circle is 9.55 (OA) and that applies to OB as well so OB is 9.55. Since we have OC we can do 9.55-8.27 to get BC which is 1.28 cm. So the perimeter is 5 + 4.75 + 1.28 = 11.03 cm
C) The area of the sector can be found by the equation Area = 0.5 x r^2 x theta
substituting in will give us 0.5 x 9.55^2 x π/6 = 23.88 cm^2. The area of the triangle can be found by 0.5 x b x h where b is base and h is height, so 0.5 x 8.27 x 4.75 and you will get 19.64. To find the area of the shaded region that's the area of the whole sector minus the area of the triangle so 23.88 - 19.64 = 4.24 cm ^2. to express it as a precentage, divide it by the area of the whole sector and multiply by 100. So the final answer will be 4.24/23.88 x 100 = 17.75%
Hope I helped <3
