1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
valkas [14]
3 years ago
11

Calculate area and perimeter​

Mathematics
1 answer:
mina [271]3 years ago
5 0

Answer:

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

You might be interested in
A sequence can be represented by the formula t(n)=6; t(n+1)=3*t(n). write the first 4 terms of the sequence. what would be an ex
Kamila [148]

Answer:

Learn how to find explicit formulas for arithmetic sequences. For example, find an explicit formula for 3, 5, 7,... ... CCSS Math: HSF. ... plug in the number of the term we are interested in, and we will get the value of that term. ... Writing explicit formulas ... Check out, for example, the following calculations of the first few terms.

Step-by-step explanation:...

8 0
3 years ago
Alvin’s age is two times Elga’s age. The sum of their age is 60. What is Elga’s age?
Nady [450]

x = Elgas age

2x = Alvins age

x +2x = 60

3x =60

x = 60/3 = 20

Elga is 20 years old

4 0
3 years ago
Why is school so boring
Elanso [62]

Answer:

because life sucks

Step-by-step explanation:

idk

5 0
3 years ago
Read 2 more answers
What is the mode of the data set 5,7,8,10,12,12
nekit [7.7K]

Answer:

The mode is twelve.

Step-by-step explanation:

The mode of a data set is the number that appears the most often. I've added an attachment so that you might be able to see some examples.

6 0
3 years ago
The time is 1:25 A.M. What time will it be in 6 hours and 55 minutes?
uranmaximum [27]

Answer:

8:20 am

Step-by-step explanation:

8 0
3 years ago
Other questions:
  • A fifth grade class wants to buy their teacher a present that costs 79.75. The students have decided to each contribute $2.75. H
    14·1 answer
  • There are 5 cards with a picture of a rose and 3 cards with a picture of a daisy. Alia keeps all the cards down on a table with
    15·1 answer
  • What is an algebraic expression for, the sum of s and t ?​
    14·1 answer
  • Please Help!! I will give brainliest to the quick and correct answer
    11·1 answer
  • vincent is going to frame a rectangular picture with the dimensions shown. write a polynomial for the perimeter.
    12·1 answer
  • Whats the answer? A B C D
    9·2 answers
  • Whats up ReaperGodofDeath which one Black butler or Black clover
    10·2 answers
  • What is the value of f(5) in f(x)=-2(x+1)
    9·1 answer
  • -19-5/4v is less than or equal too -14
    6·1 answer
  • 3m²-48m⁶<br><br>factor the polynomial completely​
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!