In the figure above, triangle ABC is inscribed in the circle with center O, such that CD is perpendicular to AB. If the length of the side AC is centimeters, andang;AOB is 60o and the smaller perimeter of the sector AOB is 12 + 2andpi; centimeters, what is the perimeter of the triangle ABC in centimeters?a)16b)c)d)e)Correct answer is option 'D'. Can you explain this answer?

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In the figure above, triangle ABC is inscribed in the circle with center O, such that CD is perpendicular to AB. If the length of the side AC is

centimeters, ∠AOB is 60^o and the smaller perimeter of the sector AOB is 12 + 2π centimeters, what is the perimeter of the triangle ABC in centimeters?

a)
16
b)
c)
d)
e)

Correct answer is option 'D'. Can you explain this answer?

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In the figure above, triangle ABC is inscribed in the circle with cent...

Given

Circle with center at O
- Let the radius of the circle be r
Triangle ABC is inscribed in the circle
- CD is perpendicular to AB
- So, OD is also perpendicular to AB. Hence AD = DB
  - This is because the perpendicular drawn from the centre of the circle to the chord of the circle bisects the chord.
AC =
centimeters
Perimeter of minor sector AOB = 12 +2π centimeters
In triangle AOB,
- ∠AOB = 60^o
- Triangle AOB is an isosceles triangle, with OA =OB =r
  - So, ∠OAB = ∠OBA
- Now, since ∠AOB = 60^o
  - ⇒ ∠OAB = ∠OBA = ∠AOB = 60^o
- Triangle AOB is an equilateral triangle
  - So, OA =OB = BA =r (All sides are equal in an equilateral triangle)
To Find: Perimeter of triangle ABC
Perimeter of triangle ABC = AC + CB + BA
We know AC = cm
So, Perimeter of triangle ABC =+ CB+BA
So, to answer the question, we need to find the length of CB and BA.

Approach

Finding length of CB
1. In triangle ABC, we have the perpendicular from vertex C bisecting the opposite base AB, hence triangle ABC should be an isosceles triangle with AC = CB.
2. As we know the length of AC, we can find the length of CB
Finding length of BA
- Length of BA is equal to r
- We know that perimeter of sector AOB =
- - 1. OA = OB = r (already established above)
    2. We are also given perimeter is equal to 12 +2π
    3. So from the above relation we will be able to calculate the value of r. which is equal to BA.
- Knowing the value of CB and BA, we will be able to find the perimeter of triangle ABC
- Working Out
- Finding length of CB
  1. We know that AC =centimetres and AC = CB.
  2. Thus CB = centimetres
  3. Finding length of BA

hence BA = 6 centimeters

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In the figure above, triangle ABC is inscribed in the circle with center O, such that CD is perpendicular to AB. If the length of the side AC is centimeters, ∠AOB is 60o and the smaller perimeter of the sector AOB is 12 + 2π centimeters, what is the perimeter of the triangle ABC in centimeters?a)16b)c)d)e)Correct answer is option 'D'. Can you explain this answer?

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