Circle and Chord

Definition of Chord

A chord is a line segment that connects two points on a circle. Unlike a diameter that passes through the center of the circle, a chord can be positioned anywhere as long as both endpoints lie on the circle.

In the figure above, $AB$, $CD$, and $EF$ are chords with different lengths.

Properties of Chords

Equal Length Chords

Two chords of equal length have the same distance from the center of the circle.

If $PQ = RS$, then the distance from center $O$ to chord $PQ$ equals the distance from $O$ to chord $RS$, that is $d_1 = d_2$.

Line from Center Perpendicular to Chord

A line drawn from the center of a circle perpendicular to a chord divides the chord into two equal parts.

In the figure above, $OM \perp AB$ and $M$ is the midpoint of chord $AB$, so $AM = MB$.

Chord Length

Chord Length Formula

To calculate the length of a chord, we can use the formula:

Where:

• $r$ = radius of the circle
• $\theta$ = central angle subtending the chord (in radians)

The relationship between central angle and chord length can be visualized as follows:

Distance of Chord from Center

The distance of a chord from the center of the circle can be calculated using the formula:

Or if the chord length $l$ is known:

Intersecting Chords Theorem

If two chords intersect inside a circle, then the product of the segments of one chord equals the product of the segments of the other chord.

In the figure above, the following holds: $PA \times PB = PC \times PD$

Inscribed Angles Subtending the Same Chord

Inscribed angles that subtend the same chord have equal measures.

In the figure above, $\angle ACB = \angle ADB$ because both subtend the same chord $AB$.

Apothem

An apothem is the shortest distance from the center of a circle to a chord, which is the perpendicular line from the center to the chord.

The length of the apothem can be calculated using the formula:

Where:

• $a$ = length of apothem
• $r$ = radius of the circle
• $l$ = length of the chord
• $\theta$ = central angle

Parallel Chords

Two parallel chords in a circle have special properties.

If $AB \parallel CD$, then arc $AC$ equals arc $BD$.

Practice Problems

1. A circle has a radius of 10 cm. If the central angle subtending a chord is 60°, determine:

   • The length of the chord
   • The distance of the chord from the center of the circle

2. Two chords $AB$ and $CD$ intersect at point $P$ inside a circle. If $PA = 4$ cm, $PB = 6$ cm, and $PC = 3$ cm, find the length of $PD$.

3. In a circle with radius 13 cm, there is a chord of length 24 cm. Calculate the distance of this chord from the center of the circle.

4. Two parallel chords in a circle are 3 cm and 4 cm away from the center respectively. If the radius of the circle is 5 cm, determine the lengths of both chords.

5. Prove that the longest chord in a circle is the diameter.

Answer Key

1. Calculating chord length and its distance from center

   Given: $r = 10$ cm, $\theta = 60° = \frac{\pi}{3}$ rad

   $$l = 2r \sin\left(\frac{\theta}{2}\right) = 2 \times 10 \times \sin\left(\frac{\pi/3}{2}\right)$$

   $$l = 20 \times \sin(30°) = 20 \times 0.5 = 10 \text{ cm}$$

   $$d = r \cos\left(\frac{\theta}{2}\right) = 10 \times \cos(30°)$$

   $$d = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \text{ cm}$$

2. Intersecting chords theorem

   Given: $PA = 4$ cm, $PB = 6$ cm, $PC = 3$ cm

   $$PA \times PB = PC \times PD$$

   $$4 \times 6 = 3 \times PD$$

   $$24 = 3 \times PD$$

   $$PD = 8 \text{ cm}$$

3. Calculating distance of chord from center

   Given: $r = 13$ cm, $l = 24$ cm

   $$d = \sqrt{r^2 - \left(\frac{l}{2}\right)^2}$$

   $$d = \sqrt{13^2 - 12^2}$$

   $$d = \sqrt{169 - 144}$$

   $$d = \sqrt{25} = 5 \text{ cm}$$

4. Parallel chords

   Given: $r = 5$ cm, $d_1 = 3$ cm, $d_2 = 4$ cm

   For the first chord:

   $$l_1 = 2\sqrt{r^2 - d_1^2} = 2\sqrt{25 - 9} = 2\sqrt{16} = 8 \text{ cm}$$

   For the second chord:

   $$l_2 = 2\sqrt{r^2 - d_2^2} = 2\sqrt{25 - 16} = 2\sqrt{9} = 6 \text{ cm}$$

5. Proof that diameter is the longest chord

   For any chord with central angle $\theta$:

   $$l = 2r \sin\left(\frac{\theta}{2}\right)$$

   The maximum value of $\sin\left(\frac{\theta}{2}\right) = 1$ is achieved when $\frac{\theta}{2} = 90°$, that is $\theta = 180°$.

   When $\theta = 180°$, the chord passes through the center of the circle (diameter) with length:

   $$l_{max} = 2r \times 1 = 2r$$

   Therefore, the diameter is the longest chord.