CAT Exam Model Paper 4 with solutions for free online practice

Question : 14 of 60

Marks: +1, -0

In the given figure AB is the diameter of a circle with center ‘O’. C is any point in the circle such that ∠ACB = 120°. OD is perpendicular to BC and the length of OD is 1.5 cm.

What is the length of AC (in cm)?

${3√3}/2$
$3√3$
$2√3$
6

Validate

Solution:

Construction: Extend BC to cut the circle at E and join AE.

∆AEB is right angled at E. (Angle in a semicircle is a right angle).
Hence, both AE and OD are perpendicular to EB. By AA property:
∆AEB -∆ODB
Hence,

{ O B }/{ A B } = { O D } /{ A E }

{ O D }/ { A E } = { 1 } /{ 2 }

⇒ AE = 2OD = 3cm
In right ∆AEC:
∠ACE = 180° ∠ACB - 180° -120° = 60°
and

A C = { A E } /{ \sin 60° } = 3 ( { 2 } /{√ { 3 } } ) = 2 √ { 3 } { \cm }

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