In the given figure, AB and CD are the longest chords of the circle with their lengths equal to 8 units. If the length of the minor arc BC is 1/6thof the perimeter of the circle, what is the length of the chord BD?a)23b)4c)43d)8e)83Correct answer is option 'C'. Can you explain this answer?

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In the given figure, AB and CD are the longest chords of the circle with their lengths equal to 8 units. If the length of the minor arc BC is 1/6th of the perimeter of the circle, what is the length of the chord BD?

a)
2√3
b)
4
c)
4√3
d)
8
e)
8√3

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

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In the given figure, AB and CD are the longest chords of the circle wi...

Given

AB = CD = 8 units and they are the longest chords of the circle.
- AB and CD are diameters of the circle.
  - As diameters are the longest chord.
- Let their intersection point be O, center of the circle.
- Radius of the circle = 4 units
Length of minor BC = 1/6∗(Perimeter of the circle)

= 1/6∗2πr

To Find: Length of chord BD?

Approach

We need to find the length of chord BD. We notice that chord BD is a part of triangle OBD.
1. In triangle OBD, we know that OB = OD = 4 units (radius of the circle)
2. The third side of this triangle, BD, is unknown that we need to find out.
Now, since OBD is an isosceles triangle with OB=OD, if we drop a perpendicular in this triangle it will bisect the BD.
1. Let’s drop a perpendicular from O to chord BD at point E.
  1. Therefore, BE = ED

Now as triangle OED is a right angled triangle, knowing one of the sides and one of the angles will be sufficient to calculate the length of ED.

In right triangle ODE,
1. We know that OD = 4 units.
  1. So, we need to find one of the angles of triangle ODE.
2. As triangle OBD is isosceles and OE is perpendicular from O, OE will bisect ∠BOD. So, ∠DOE = ∠BOD/2
3. Also, ∠BOD = 180 - ∠COB. So, if we can calculate ∠COB, we can find the value of ∠BOD
We know that length of arc BC =
We can use the above relation to calculate the value of ∠COB

Working Out

1. Length of arc

which gives us ∠COB = 60^o

4. As triangle ODE is a 30^o - 60^o -90^o triangle, we have the length of ED = 2√3

5. Thus BD = 2 * 2√3 = 4√3

Answer: C

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In the given figure, AB and CD are the longest chords of the circle with their lengths equal to 8 units. If the length of the minor arc BC is 1/6thof the perimeter of the circle, what is the length of the chord BD?a)2√3b)4c)4√3d)8e)8√3Correct answer is option 'C'. Can you explain this answer?

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