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# In the given figure AB is the diameter of the circle, whose center is

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VP
Joined: 19 Oct 2018
Posts: 1171
Location: India
In the given figure AB is the diameter of the circle, whose center is  [#permalink]

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27 Nov 2019, 08:53
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Difficulty:

55% (hard)

Question Stats:

56% (02:45) correct 44% (03:10) wrong based on 16 sessions

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In the given figure AB is the diameter of the circle, whose center is O. Point C lies inside the circle such that ∠ACB=120 degrees. Also, OD is perpendicular to BC and length of OD is 1.5 cm. Find the length of AC?

A. $$\frac{3\sqrt{3}}{2}$$
B. 3
C. $$2\sqrt{3}$$
D. $$3\sqrt{3}$$
E. 6

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Math Expert
Joined: 02 Aug 2009
Posts: 8305
In the given figure AB is the diameter of the circle, whose center is  [#permalink]

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27 Nov 2019, 21:13
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nick1816 wrote:
In the given figure AB is the diameter of the circle, whose center is O. Point C lies inside the circle such that ∠ACB=120 degrees. Also, OD is perpendicular to BC and length of OD is 1.5 cm. Find the length of AC?

A. $$\frac{3\sqrt{3}}{2}$$
B. 3
C. $$2\sqrt{3}$$
D. $$3\sqrt{3}$$
E. 6

Extend BC to meet the circle at E, and join AE.
Now AE||OD and double of OD, since BOD and ABE are similar triangles in the ratio 1:2=BO:BA.

Take triangle ACE.
Angle AEB= angle AEC = 90, as it is opposite the diameter AB.
Angle AEC=180-120=60.

Triangle AEC thus is 30-60-90 triangle.
Now AE opposite 60 degree angle is 3, so hypotenuse $$AC=\frac{2*3}{\sqrt{3}}=2\sqrt{3}$$, as the sides are in ratio 1:sqrt3:2

C
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In the given figure AB is the diameter of the circle, whose center is   [#permalink] 27 Nov 2019, 21:13
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