Question

What is the diameter of the circle in the figure ? 

 

• A. 2√(3) centimetres
• B. 4 centimetres
• C. 3 centimetres
• D. 4√(3) centimetres

Hint:

For a 30-60-90 right-angled triangle, the sides are in the ratio 1 : √(3) : 2. The side opposite the 30° angle is the shortest (1), the side opposite the 60° angle is √(3) times the shortest, and the hypotenuse (opposite the 90° angle) is twice the shortest side. Here, the side opposite 30° is 2, so the hypotenuse (diameter) must be 2 × 2 = 4.

Answer 1

The Correct Option is B

The figure shows a triangle inscribed in a circle. One of the sides of the triangle is the diameter of the circle. We are given one angle (30°) and the length of the side opposite to it (2 cm). We need to find the diameter.

There are two main approaches:
1. Geometric Property:
The angle subtended by a diameter at any point on the circumference is a right angle (90°). This means the triangle is a right-angled triangle with the diameter as its hypotenuse.
2. Trigonometry:
In a right-angled triangle, we can use trigonometric ratios (sin, cos, tan). Specifically, (θ) = OppositeHypotenuse.

Let the vertices of the triangle be A, B, and C. Let AC be the diameter. The angle at vertex B on the circumference is ABC = 90^.
We are given BAC = 30^ and the side opposite to it, BC = 2 cm.
The diameter AC is the hypotenuse of the right-angled triangle ABC.
Using the sine ratio for angle A:
(A) = Opposite sideHypotenuse (30^) = (BC)/(AC) We know that (30^) = (1)/(2) and BC = 2 cm.
(1)/(2) = (2)/(AC) Now, we solve for AC (the diameter):
AC = 2 × 2 = 4 cm The diameter of the circle is 4 centimetres.