Question: 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?
- \(\frac{3√3}{2}\)
- 3
- 2√3
- 3√3
- 6
Correct Answer: C
Solution and Explanation:
Approach Solution 1:
The problem statement states that:
Given:
- 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.
- OD is perpendicular to BC and the length of OD is 1.5 cm.
Find out:
- The length of AC.
Let’s extend BC to meet the circle at E, and join AE.
∠BEA is 90 as it is an angle on the diameter AB.
Therefore, AE||OD, as angle AEB and ODB are 90
AE is 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
Therefore, hypotenuse AC= 2∗3/√3 =2√3, as the sides are in ratio 1:√3:2
Hence the length of AC = 2√3.
Approach Solution 2:
The problem statement informs that:
Given:
- 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.
- OD is perpendicular to BC and the length of OD is 1.5 cm.
Find out:
- The length of AC.
We know that ∠ACB = 120 degrees. We can use the law of cosines to find the length of AC.
AC^2 = AB^2 + BC^2 - 2 * AB * BC * cos(∠ACB)
AB = 2 * OD = 2 * 1.5 = 3
BC = √(AC^2 - AB^2 + 2 * AB * BC * cos(∠ACB))
Substituting the values in the above equation, we get:
AC^2 = 3^2 + BC^2 - 2 * 3 * BC * cos(120)
Solving for AC, we get:
AC = √(3^2 + BC^2 - 2 * 3 * BC * -0.5) = 2√3.
Hence the length of AC = 2√3.
Approach Solution 3:
The problem statement implies that:
Given:
- 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.
- OD is perpendicular to BC and the length of OD is 1.5 cm.
Find out:
- The length of AC.
We know that ∠ACB = 120 degrees and AB is the diameter of the circle, so ∠BOC = 60 degrees.
Using the property of inscribed angles, we know that:
∠BOC = ∠BAC/2
∠BAC = 2 * ∠BOC = 2 * 60 = 120 degrees
Since ∠BAC = ∠ACB, we know that ∠ACB = 120 degrees.
Using the law of sines, we can find the length of AC.
AC/sin(∠ACB) = AB/sin(∠BAC)
AC/sin(120) = AB/sin(120)
AC = AB/sin(120) = 3/(√3/2) — [where AB = 2*OD = 2 x 1.5 = 3]
AC = 6/√3 = 6√3/3 = 2√3
Hence the length of AC = 2√3.
“In the given figure AB is the diameter of the circle, whose center is O”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This book has been taken from the book “GMAT Official Guide 2021”. To solve the GMAT Problem Solving questions, the candidates must have a basic concept of calculations and mathematics. The candidates can analyse GMAT Quant practice papers to practice varieties of questions that will enable them to improve their mathematical skills.
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