# Nakafa Learning Content
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URL: https://nakafa.com/en/subjects/mathematics/circle/circle-and-chord
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/circle/circle-and-chord/en.mdx
Explore circle and chord relationships. Understand theorems, properties, and solve chord length problems with clear examples and worked solutions.
---
## 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.
Component: LineEquation
Props:
- title: Chords on a Circle
- description: Various chords with different lengths on a circle.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [{ x: 0, y: 0, z: 0 }],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "O", at: 0, offset: [-0.3, -0.3, 0] }],
},
{
points: (() => {
const angle1 = Math.PI / 6;
const angle2 = Math.PI / 3;
return [
{ x: 3 * Math.cos(angle1), y: 3 * Math.sin(angle1), z: 0 },
{ x: 3 * Math.cos(angle2), y: 3 * Math.sin(angle2), z: 0 },
];
})(),
color: getColor("CYAN"),
showPoints: true,
labels: [
{ text: "A", at: 0, offset: [0.3, 0.3, 0] },
{ text: "B", at: 1, offset: [0.3, 0.3, 0] },
],
},
{
points: (() => {
const angle1 = (2 * Math.PI) / 3;
const angle2 = (5 * Math.PI) / 6;
return [
{ x: 3 * Math.cos(angle1), y: 3 * Math.sin(angle1), z: 0 },
{ x: 3 * Math.cos(angle2), y: 3 * Math.sin(angle2), z: 0 },
];
})(),
color: getColor("TEAL"),
showPoints: true,
labels: [
{ text: "C", at: 0, offset: [-0.3, 0.3, 0] },
{ text: "D", at: 1, offset: [-0.3, 0.3, 0] },
],
},
{
points: (() => {
const angle1 = (7 * Math.PI) / 6;
const angle2 = (3 * Math.PI) / 2;
return [
{ x: 3 * Math.cos(ang ... [truncated; 1459 chars]
- cameraPosition: [0, 0, 10]
- showZAxis: false
In the figure above, $$AB$$, $$CD$$, and $$EF$$ are chords with different lengths.
Visible text: In the figure above, , , and 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.
Component: LineEquation
Props:
- title: Equal Length Chords
- description: Two chords with equal length have the same distance from the center.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 4 * Math.cos(angle),
y: 4 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [{ x: 0, y: 0, z: 0 }],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "O", at: 0, offset: [-0.3, -0.3, 0] }],
},
{
points: (() => {
const y = 2;
const x = Math.sqrt(16 - y * y);
return [
{ x: -x, y: y, z: 0 },
{ x: x, y: y, z: 0 },
];
})(),
color: getColor("CYAN"),
showPoints: true,
labels: [
{ text: "P", at: 0, offset: [-0.3, 0.3, 0] },
{ text: "Q", at: 1, offset: [0.3, 0.3, 0] },
],
},
{
points: (() => {
const y = -2;
const x = Math.sqrt(16 - y * y);
return [
{ x: -x, y: y, z: 0 },
{ x: x, y: y, z: 0 },
];
})(),
color: getColor("TEAL"),
showPoints: true,
labels: [
{ text: "R", at: 0, offset: [-0.3, -0.3, 0] },
{ text: "S", at: 1, offset: [0.3, -0.3, 0] },
],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 0, y: 2, z: 0 },
],
color: getColor("AMBER"),
showPoints: false,
smooth: false,
labels: [{ text: "d₁", at: 1, offset: [0.5, 0, 0] }],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 0, y: -2, z: 0 },
],
color: getColor("AMBER"),
showPoints: false,
smooth: false, ... [truncated; 1258 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false
If $$PQ = RS$$, then the distance from center $$O$$ to chord $$PQ$$ equals the distance from $$O$$ to chord $$RS$$, that is $$d_1 = d_2$$.
Visible text: If , then the distance from center to chord equals the distance from to chord , that is .
### 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.
Component: LineEquation
Props:
- title: Perpendicular Line from Center
- description: A line from the center perpendicular to a chord bisects it.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 5 * Math.cos(angle),
y: 5 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [{ x: 0, y: 0, z: 0 }],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "O", at: 0, offset: [-0.3, -0.3, 0] }],
},
{
points: (() => {
const angle1 = Math.PI / 4;
const angle2 = (3 * Math.PI) / 4;
return [
{ x: 5 * Math.cos(angle1), y: 5 * Math.sin(angle1), z: 0 },
{ x: 5 * Math.cos(angle2), y: 5 * Math.sin(angle2), z: 0 },
];
})(),
color: getColor("CYAN"),
showPoints: true,
labels: [
{ text: "A", at: 0, offset: [0.3, 0.3, 0] },
{ text: "B", at: 1, offset: [-0.3, 0.3, 0] },
],
},
{
points: (() => {
const midAngle = (Math.PI / 4 + (3 * Math.PI) / 4) / 2;
const midX = 5 * Math.cos(midAngle);
const midY = 5 * Math.sin(midAngle);
const chordMidX =
(5 * Math.cos(Math.PI / 4) + 5 * Math.cos((3 * Math.PI) / 4)) / 2;
const chordMidY =
(5 * Math.sin(Math.PI / 4) + 5 * Math.sin((3 * Math.PI) / 4)) / 2;
return [
{ x: 0, y: 0, z: 0 },
{ x: chordMidX, y: chordMidY, z: 0 },
];
})(),
color: getColor("PINK"),
showPoints: true,
smooth: false,
labels: [{ text: "M", at: 1, of ... [truncated; 1690 chars]
- cameraPosition: [0, 0, 15]
- showZAxis: false
In the figure above, $$OM \perp AB$$ and $$M$$ is the midpoint of chord $$AB$$, so $$AM = MB$$.
Visible text: In the figure above, and is the midpoint of chord , so .
## Chord Length
### Chord Length Formula
To calculate the length of a chord, we can use the formula:
```math
AB = 2r \sin\left(\frac{\theta}{2}\right)
```
Where:
- $$r$$ = radius of the circle
- $$\theta$$ = central angle subtending the chord (in radians)
Visible text: - = radius of the circle
- = central angle subtending the chord (in radians)
The relationship between central angle and chord length can be visualized as follows:
Component: LineEquation
Props:
- title: Central Angle and Chord
- description: Relationship between central angle and chord length.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 4 * Math.cos(angle),
y: 4 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [{ x: 0, y: 0, z: 0 }],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "O", at: 0, offset: [-0.3, -0.3, 0] }],
},
{
points: (() => {
const angle1 = Math.PI / 6;
const angle2 = (2 * Math.PI) / 3;
return [
{ x: 4 * Math.cos(angle1), y: 4 * Math.sin(angle1), z: 0 },
{ x: 4 * Math.cos(angle2), y: 4 * Math.sin(angle2), z: 0 },
];
})(),
color: getColor("CYAN"),
showPoints: true,
labels: [
{ text: "P", at: 0, offset: [0.3, 0.3, 0] },
{ text: "Q", at: 1, offset: [-0.3, 0.3, 0] },
],
},
{
points: (() => {
const angle1 = Math.PI / 6;
return [
{ x: 0, y: 0, z: 0 },
{ x: 4 * Math.cos(angle1), y: 4 * Math.sin(angle1), z: 0 },
];
})(),
color: getColor("TEAL"),
showPoints: false,
smooth: false,
},
{
points: (() => {
const angle2 = (2 * Math.PI) / 3;
return [
{ x: 0, y: 0, z: 0 },
{ x: 4 * Math.cos(angle2), y: 4 * Math.sin(angle2), z: 0 },
];
})(),
color: getColor("TEAL"),
showPoints: false,
smooth: false,
},
{
points: (() => {
const startAngle = Math.PI / 6;
const endAn ... [truncated; 1558 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false
### Distance of Chord from Center
The distance of a chord from the center of the circle can be calculated using the formula:
```math
d = r \cos\left(\frac{\theta}{2}\right)
```
Or if the chord length $$l$$ is known:
Visible text: Or if the chord length is known:
```math
d = \sqrt{r^2 - \left(\frac{l}{2}\right)^2}
```
## 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.
Component: LineEquation
Props:
- title: Intersection of Two Chords
- description: Intersecting chords theorem: $$PA \times PB = PC \times PD$$.
Visible text: Intersecting chords theorem: .
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 5 * Math.cos(angle),
y: 5 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: (() => {
const angle1 = Math.PI / 3;
const angle2 = (4 * Math.PI) / 3;
return [
{ x: 5 * Math.cos(angle1), y: 5 * Math.sin(angle1), z: 0 },
{ x: 5 * Math.cos(angle2), y: 5 * Math.sin(angle2), z: 0 },
];
})(),
color: getColor("CYAN"),
showPoints: true,
labels: [
{ text: "A", at: 0, offset: [0.3, 0.3, 0] },
{ text: "B", at: 1, offset: [-0.3, -0.3, 0] },
],
},
{
points: (() => {
const angle1 = (5 * Math.PI) / 6;
const angle2 = (11 * Math.PI) / 6;
return [
{ x: 5 * Math.cos(angle1), y: 5 * Math.sin(angle1), z: 0 },
{ x: 5 * Math.cos(angle2), y: 5 * Math.sin(angle2), z: 0 },
];
})(),
color: getColor("TEAL"),
showPoints: true,
labels: [
{ text: "C", at: 0, offset: [-0.3, 0.3, 0] },
{ text: "D", at: 1, offset: [0.3, -0.3, 0] },
],
},
{
points: (() => {
// Calculate intersection point P
const A = {
x: 5 * Math.cos(Math.PI / 3),
y: 5 * Math.sin(Math.PI / 3),
};
const B = {
x: 5 * Math.cos((4 * Math.PI) / 3),
y: 5 * Math.sin((4 * Math.PI) / 3),
};
const C = {
x: 5 * Math.cos((5 * ... [truncated; 1794 chars]
- cameraPosition: [0, 0, 15]
- showZAxis: false
In the figure above, the following holds: $$PA \times PB = PC \times PD$$
Visible text: In the figure above, the following holds:
## Inscribed Angles Subtending the Same Chord
Inscribed angles that subtend the same chord have equal measures.
Component: LineEquation
Props:
- title: Inscribed Angles Subtending the Same Chord
- description: Inscribed angles subtending chord $$AB$$ have equal
measures.
Visible text: Inscribed angles subtending chord have equal
measures.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 4 * Math.cos(angle),
y: 4 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: (() => {
const angleA = 0;
const angleB = Math.PI / 2;
return [
{ x: 4 * Math.cos(angleA), y: 4 * Math.sin(angleA), z: 0 },
{ x: 4 * Math.cos(angleB), y: 4 * Math.sin(angleB), z: 0 },
];
})(),
color: getColor("CYAN"),
showPoints: true,
labels: [
{ text: "A", at: 0, offset: [0.5, 0, 0] },
{ text: "B", at: 1, offset: [0, 0.5, 0] },
],
},
{
points: (() => {
const angleC = (3 * Math.PI) / 4;
const angleA = 0;
const angleB = Math.PI / 2;
return [
{ x: 4 * Math.cos(angleC), y: 4 * Math.sin(angleC), z: 0 },
{ x: 4 * Math.cos(angleA), y: 4 * Math.sin(angleA), z: 0 },
{ x: 4 * Math.cos(angleB), y: 4 * Math.sin(angleB), z: 0 },
{ x: 4 * Math.cos(angleC), y: 4 * Math.sin(angleC), z: 0 },
];
})(),
color: getColor("PINK"),
showPoints: true,
smooth: false,
labels: [{ text: "C", at: 0, offset: [-0.3, 0.3, 0] }],
},
{
points: (() => {
const angleD = (5 * Math.PI) / 4;
const angleA = 0;
const angleB = Math.PI / 2;
return [
{ x: 4 * Math.cos(angleD), y: 4 * Math.sin(angleD), z: 0 },
{ x: ... [truncated; 1505 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false
In the figure above, $$\angle ACB = \angle ADB$$ because both subtend the same chord $$AB$$.
Visible text: In the figure above, because both subtend the same chord .
## 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.
Component: LineEquation
Props:
- title: Apothem of a Chord
- description: Apothem is the perpendicular line from the center to the chord.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 5 * Math.cos(angle),
y: 5 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [{ x: 0, y: 0, z: 0 }],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "O", at: 0, offset: [-0.3, -0.3, 0] }],
},
{
points: (() => {
const y = 3;
const x = Math.sqrt(25 - y * y);
return [
{ x: -x, y: y, z: 0 },
{ x: x, y: y, z: 0 },
];
})(),
color: getColor("CYAN"),
showPoints: true,
labels: [
{ text: "P", at: 0, offset: [-0.3, 0.3, 0] },
{ text: "Q", at: 1, offset: [0.3, 0.3, 0] },
],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 0, y: 3, z: 0 },
],
color: getColor("PINK"),
showPoints: true,
smooth: false,
labels: [
{ text: "M", at: 1, offset: [0.5, 0, 0] },
{ text: "apothem", at: 0.5, offset: [1.2, 0, 0] },
],
},
{
points: [
{ x: -0.3, y: 3, z: 0 },
{ x: -0.3, y: 2.7, z: 0 },
{ x: 0, y: 2.7, z: 0 },
],
color: getColor("AMBER"),
showPoints: false,
smooth: false,
},
]
- cameraPosition: [0, 0, 15]
- showZAxis: false
The length of the apothem can be calculated using the formula:
```math
a = r \cos\left(\frac{\theta}{2}\right) = \sqrt{r^2 - \left(\frac{l}{2}\right)^2}
```
Where:
- $$a$$ = length of apothem
- $$r$$ = radius of the circle
- $$l$$ = length of the chord
- $$\theta$$ = central angle
Visible text: - = length of apothem
- = radius of the circle
- = length of the chord
- = central angle
## Parallel Chords
Two parallel chords in a circle have special properties.
Component: LineEquation
Props:
- title: Parallel Chords
- description: Arcs between parallel chords have equal lengths.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 4 * Math.cos(angle),
y: 4 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: (() => {
const y = 2;
const x = Math.sqrt(16 - y * y);
return [
{ x: -x, y: y, z: 0 },
{ x: x, y: y, z: 0 },
];
})(),
color: getColor("CYAN"),
showPoints: true,
labels: [
{ text: "A", at: 0, offset: [-0.3, 0.3, 0] },
{ text: "B", at: 1, offset: [0.3, 0.3, 0] },
],
},
{
points: (() => {
const y = -2;
const x = Math.sqrt(16 - y * y);
return [
{ x: -x, y: y, z: 0 },
{ x: x, y: y, z: 0 },
];
})(),
color: getColor("TEAL"),
showPoints: true,
labels: [
{ text: "C", at: 0, offset: [-0.3, -0.3, 0] },
{ text: "D", at: 1, offset: [0.3, -0.3, 0] },
],
},
{
points: (() => {
const y1 = 2;
const x1 = Math.sqrt(16 - y1 * y1);
const angle1 = Math.atan2(y1, -x1);
const y2 = -2;
const x2 = Math.sqrt(16 - y2 * y2);
const angle2 = Math.atan2(y2, -x2);
const numPoints = 20;
return Array.from({ length: numPoints + 1 }, (_, i) => {
const angle = angle1 + (angle2 - angle1) * (i / numPoints);
return {
x: 4 * Math.cos(angle),
y: 4 * Math.sin(angle),
z: 0,
};
});
})(),
color: getColor("PINK"),
sho ... [truncated; 1673 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false
If $$AB \parallel CD$$, then arc $$AC$$ equals arc $$BD$$.
Visible text: If , then arc equals arc .
## Practice Problems
1. A circle has a radius of $$10 \text{ cm}$$. If the central angle subtending a chord is $$60^\circ$$, 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 $$PA = 4 \text{ cm}$$, $$PB = 6 \text{ cm}$$, and $$PC = 3 \text{ cm}$$, find the length of $$PD$$.
3. In a circle with radius $$13 \text{ cm}$$, there is a chord of length $$24 \text{ cm}$$. Calculate the distance of this chord from the center of the circle.
4. Two parallel chords in a circle are $$3 \text{ cm}$$ and $$4 \text{ cm}$$ away from the center respectively. If the radius of the circle is $$5 \text{ cm}$$, determine the lengths of both chords.
5. Prove that the longest chord in a circle is the diameter.
Visible text: 1. A circle has a radius of . If the central angle subtending a chord is , determine:
- The length of the chord
- The distance of the chord from the center of the circle
2. Two chords and intersect at point inside a circle. If , , and , find the length of .
3. In a circle with radius , there is a chord of length . Calculate the distance of this chord from the center of the circle.
4. Two parallel chords in a circle are and away from the center respectively. If the radius of the circle is , 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: $$r = 10 \text{ cm}$$, $$\theta = 60^\circ = \frac{\pi}{3}$$ rad
```math
l = 2r \sin\left(\frac{\theta}{2}\right) = 2 \times 10 \times \sin\left(\frac{\pi/3}{2}\right)
```
```math
l = 20 \times \sin(30^\circ) = 20 \times 0.5 = 10 \text{ cm}
```
```math
d = r \cos\left(\frac{\theta}{2}\right) = 10 \times \cos(30^\circ)
```
```math
d = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \text{ cm}
```
2. **Intersecting chords theorem**
Given: $$PA = 4 \text{ cm}$$, $$PB = 6 \text{ cm}$$, $$PC = 3 \text{ cm}$$
```math
PA \times PB = PC \times PD
```
```math
4 \times 6 = 3 \times PD
```
```math
24 = 3 \times PD
```
```math
PD = 8 \text{ cm}
```
3. **Calculating distance of chord from center**
Given: $$r = 13 \text{ cm}$$, $$l = 24 \text{ cm}$$
```math
d = \sqrt{r^2 - \left(\frac{l}{2}\right)^2}
```
```math
d = \sqrt{13^2 - 12^2}
```
```math
d = \sqrt{169 - 144}
```
```math
d = \sqrt{25} = 5 \text{ cm}
```
4. **Parallel chords**
Given: $$r = 5 \text{ cm}$$, $$d_1 = 3 \text{ cm}$$, $$d_2 = 4 \text{ cm}$$
For the first chord:
```math
l_1 = 2\sqrt{r^2 - d_1^2} = 2\sqrt{25 - 9} = 2\sqrt{16} = 8 \text{ cm}
```
For the second chord:
```math
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 $$\theta$$:
```math
l = 2r \sin\left(\frac{\theta}{2}\right)
```
The maximum value of $$\sin\left(\frac{\theta}{2}\right) = 1$$ is achieved when $$\frac{\theta}{2} = 90^\circ$$, that is $$\theta = 180^\circ$$.
When $$\theta = 180^\circ$$, the chord passes through the center of the circle (diameter) with length:
```math
l_{max} = 2r \times 1 = 2r
```
Therefore, the diameter is the longest chord.
Visible text: 1. **Calculating chord length and its distance from center**
Given: , rad
2. **Intersecting chords theorem**
Given: , ,
3. **Calculating distance of chord from center**
Given: ,
4. **Parallel chords**
Given: , ,
For the first chord:
For the second chord:
5. **Proof that diameter is the longest chord**
For any chord with central angle :
The maximum value of is achieved when , that is .
When , the chord passes through the center of the circle (diameter) with length:
Therefore, the diameter is the longest chord.