area ≈ 12.505
perimeter ≈ 16.1684
Step-by-step explanation:
We are given
- the radius of the circle (and therefore area of the circle)
- the area of the triangle
We want to find
- angle AOB/AOT. We want to find this because 360/the angle gives us how many OABs fit into the circle. For example, if AOT was 30 degrees, 360/30 = 12 (there are 360 degrees in a circle, so that's where 360 comes from). The area of the circle is equal to πr² = π6² = 36π, and because AOT is 30 degrees, there are 12 equal parts of sector OAB in the circle, so 36π/12=3π would be the area of the sector. A similar conclusion can be reached from the circumference instead of the area to find the distance between A and B along the circle, and OA + AB + BO = the perimeter of the minor sector.
First, we can say that OAT is a right triangle because a tangent line is perpendicular to the line from the center to the point on the circle, so AT is perpendicular to OA. This forms two right angles, one of which is OAT
One thing that we can start to solve is AT. We know that the area of a triangle is equal to base * height /2, and the height of this triangle is AO, with the base being AT. Therefore, we can say
15 = AO * AT / 2
15 = 6 * AT / 2
15 = 3 * AT
divide both sides by 3 to isolate AT
AT = 5
Because OAT is a right triangle, we can say that the hypotenuse ² = the sum of the squares of the two other lengths. The hypotenuse is opposite of the largest angle (in this case, the right angle, as in a right triangle, the right angle is always the largest), so it is OT in this case. The other two sides are OA and AT, so we can say that
OA² + AT² = OT²
5²+6² = OT²
25+36=61=OT²
square root both sides
OT = √61
Next, the Law of Sines states that
sinA/a = sinB/b = sinC/c with angles A, B, and C with sides a, b, and c. Corresponding sides are opposite their corresponding angles, so in this case, AT corresponds to angle AOT, OT corresponds to angle OAT, and AO corresponds to angle ATO.
We want to find angle AOT, as stated earlier, so we have
sin(OAT)/OT = sin(ATO)/OA = sin(AOT)/AT
We know the side lengths as well as OAT/sin(OAT) and want to figure out AOT/sin(AOT), so one equation that helps us get there is
sin(OAT)/OT = sin(AOT)/AT, encompassing our 3 known values and isolating the one unknown. We thus have
sin(90)/√61 = sin(AOT) /5
plug in sin(90) = 1
1/√61 = sin(AOT)/5
multiply both sides by 5 to isolate sin(AOT)
5/√61 = sin(AOT)
we can thus say that
arcsin(5/√61) = AOT ≈39.80557
As stated previously, given ∠AOT, we can find the area and perimeter of the sector. There are 360/39.80557 ≈ 9.04396 equal parts of sector OAB in the circle. The area of the circle is πr² = 36π, so 36π / 9.04396 ≈ 12.505 as the area. The circumference is equal to π * diameter = π * 2 * radius = 12 * π, and there are 9.04396 equal parts of arc AB in the circumference, so the length of arc is 12π / 9.04396 ≈ 4.1684. Add that to OA and OB (both are equal to the radius of 6, as any point from the center to a point on the circle is equal to the radius) to get 6+6 + 4.1684 = 16.1684 as the perimeter of the sector
