Finding Chord Length in a Unit Circle

Let's find the length of a chord in a unit circle when the angle subtended at the center is known. A unit circle is a circle with a radius of 1 unit.

Consider a unit circle with center O and radius R = 1. Let AB be the chord of this circle, and the angle subtended by the chord at the center is \(\theta\). So, \(\angle AOB = \theta\).

To find the length of the chord AB, we can draw a line segment from the center O perpendicular to the chord AB. Let this perpendicular meet the chord at point M.

• The perpendicular from the center to a chord bisects the chord. This means AM = MB, and AB = 2 * AM.
• The perpendicular from the center to a chord also bisects the angle subtended by the chord at the center. This means \(\angle AOM = \angle BOM = \frac{\theta}{2}\).

Now, consider the right-angled triangle \(\triangle OMA\).

• The hypotenuse is OA, which is the radius of the unit circle, so OA = 1.
• The angle \(\angle AOM = \frac{\theta}{2}\).
• AM is the side opposite to the angle \(\angle AOM\).

In a right-angled triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. Using trigonometry in \(\triangle OMA\):

\(\sin(\angle AOM) = \frac{AM}{OA}\)

Substitute the values:

\(\sin\left(\frac{\theta}{2}\right) = \frac{AM}{1}\)

So, the length of AM is:

\(AM = \sin\left(\frac{\theta}{2}\right)\)

The length of the chord AB is twice the length of AM:

\(AB = 2 \times AM\)

\(AB = 2 \sin\left(\frac{\theta}{2}\right)\)

Thus, the length of the chord of a unit circle which subtends an angle \(\theta\) at the center is \(2\sin \left( \frac{\theta }{2} \right)\).

Comparing with Options

Let's compare our result with the given options:

1. \(\sin \left( \frac{\theta }{2} \right)\)
2. \(\cos \left( \frac{\theta }{2} \right)\)
3. \(2\sin \left( \frac{\theta }{2} \right)\)
4. \(2\cos \left( \frac{\theta }{2} \right)\)

Our derived chord length \(2\sin \left( \frac{\theta }{2} \right)\) matches option 3.

Summary of Chord Length Calculation

Here is a summary of the steps:

1. Identify the radius of the unit circle (R=1).
2. Identify the angle subtended at the center (\(\theta\)).
3. Understand that the perpendicular from the center bisects the chord and the central angle.
4. Form a right-angled triangle using the radius, half-chord, and the perpendicular.
5. Use trigonometry (specifically sine) in the right triangle to find half the chord length.
6. Double the half-chord length to get the full chord length.

Revision Table: Key Geometry Concepts

Additional Information: General Chord Length Formula

For any circle with radius R (not just a unit circle) and a chord subtending an angle \(\theta\) at the center, the length of the chord can be found using the same method. In the right-angled triangle formed by the radius (R), half the chord (c/2), and the perpendicular from the center:

\(\sin\left(\frac{\theta}{2}\right) = \frac{\text{Half Chord}}{\text{Radius}} = \frac{c/2}{R}\)

So, \(\frac{c}{2} = R \sin\left(\frac{\theta}{2}\right)\).

The full chord length \(c = 2 R \sin\left(\frac{\theta}{2}\right)\).

In our specific problem, since it is a unit circle, R=1, which simplifies the formula to \(c = 2 \times 1 \times \sin\left(\frac{\theta}{2}\right) = 2 \sin\left(\frac{\theta}{2}\right)\).