Understanding the Geometry Problem

The question asks for the angle subtended by a chord at the centre of a circle, given the chord's length and the circle's diameter. Let's break down the given information:

• Length of the chord AB = 10 cm.
• Diameter of the circle = 20 cm.

We need to find the angle formed by the chord AB at the centre of the circle. Let O be the centre of the circle.

Calculating the Circle's Radius

The radius of a circle is half of its diameter. Given the diameter is 20 cm, the radius (let's call it 'r') is:

$\text{r} = \frac{\text{Diameter}}{2}$

$\text{r} = \frac{20 \text{ cm}}{2}$

$\text{r} = 10 \text{ cm}$

So, the radius of the circle is 10 cm.

Forming a Triangle at the Centre

Consider the triangle formed by the two radii connecting the endpoints of the chord to the centre, and the chord itself. In our case, this is triangle AOB, where O is the centre, and A and B are the endpoints of the chord AB.

• OA is a radius, so OA = 10 cm.
• OB is a radius, so OB = 10 cm.
• AB is the chord, and its length is given as 10 cm.

So, the sides of triangle AOB are OA = 10 cm, OB = 10 cm, and AB = 10 cm.

Identifying the Type of Triangle AOB

In triangle AOB, we observe that all three sides are equal in length:

OA = OB = AB = 10 cm

A triangle with all three sides equal in length is known as an equilateral triangle.

Determining the Angles of Triangle AOB

A key property of equilateral triangles is that all three internal angles are equal. The sum of angles in any triangle is 180 degrees.

Let $\angle\text{AOB}$, $\angle\text{OAB}$, and $\angle\text{OBA}$ be the angles of triangle AOB.

Since it is an equilateral triangle, $\angle\text{AOB} = \angle\text{OAB} = \angle\text{OBA}$.

The sum of angles = $\angle\text{AOB} + \angle\text{OAB} + \angle\text{OBA} = 180^\circ$.

Let $\angle\text{AOB} = \theta$. Then $\theta + \theta + \theta = 180^\circ$.

$3\theta = 180^\circ$

$\theta = \frac{180^\circ}{3}$

$\theta = 60^\circ$

Thus, each angle in triangle AOB is 60 degrees.

The Angle Subtended at the Centre

The angle subtended by the chord AB at the centre O is $\angle\text{AOB}$. We found that $\angle\text{AOB} = 60^\circ$.

Analysis of Options

Let's look at the given options for the angle subtended by the chord AB at the centre:

Based on our calculation, the angle subtended by the chord AB at the centre is 60 degrees.

Revision Table: Circle Chord Angle

Additional Information on Circle Geometry

Understanding chords and the angles they subtend is fundamental in circle geometry. Here are some related concepts:

• Angle at the Circumference: The angle subtended by a chord at any point on the circumference is half the angle subtended by the same chord at the centre, provided both angles are on the same side of the chord. In this case, the angle subtended at the circumference would be $60^\circ / 2 = 30^\circ$.
• Perpendicular from Centre: A perpendicular drawn from the centre of a circle to a chord bisects the chord. If we draw a perpendicular from O to AB, it would meet AB at its midpoint, say M. Then AM = MB = 5 cm. Triangle OMA would be a right-angled triangle.
• Congruent Chords: Equal chords of a circle subtend equal angles at the centre. Conversely, if two chords subtend equal angles at the centre, they are equal in length.

These properties are useful for solving various problems involving chords and angles in circles.