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URL: https://nakafa.com/en/subjects/mathematics/circle-arc-sector/chord
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/circle-arc-sector/chord/en.mdx
Explore circle chords and their distance-length relationships. Learn chord length formulas, intersecting theorems, and solve challenging geometry problems.
---
## Understanding Chord
A chord is a straight line segment that connects two points on a circle. Imagine it like a string stretched inside a circle, connecting two points on the edge of the circle. Each chord divides the circle into two parts, namely the minor arc and the major arc.
Chords have special characteristics in a circle. The farther the chord is from the center of the circle, the shorter its length. Conversely, the longest chord is the one that passes through the center of the circle, which is called the diameter.
Component: LineEquation
Props:
- title: Chord Visualization in Circle
- description: Various chords with different lengths and positions.
- cameraPosition: [0, 0, 12]
- showZAxis: false
- data: [
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points: createCircleOutlinePoints(4),
color: getColor("CYAN"),
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showPoints: false,
labels: [{ text: "Circle", at: 25, offset: [1, 0.5, 0] }]
},
{
points: createCircleChordPoints({
radius: 4,
startDegrees: 30,
sweepDegrees: 120,
}),
color: getColor("ORANGE"),
smooth: false,
showPoints: true,
labels: [
{ text: "A", at: 0, offset: [0.3, 0.3, 0] },
{ text: "B", at: 1, offset: [-0.3, 0.3, 0] }
]
},
{
points: createCircleChordPoints({
radius: 4,
startDegrees: 60,
sweepDegrees: 180,
}),
color: getColor("PURPLE"),
smooth: false,
showPoints: true,
labels: [
{ text: "C", at: 0, offset: [0.3, 0.3, 0] },
{ text: "D", at: 1, offset: [-0.3, -0.3, 0] }
]
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labels: [
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labels: [{ text: "O", at: 0, offset: [0.3, 0.3, 0] }]
}
]
## Types of Chords
Based on their position and characteristics, chords can be distinguished into several types:
1. **Regular Chord** is a chord that does not pass through the center of the circle. Its length is always shorter than the diameter of the circle.
2. **Diameter** is a special chord that passes through the center of the circle. The diameter is the longest chord in the circle and divides the circle into two equal parts.
3. **Parallel Chords** are two or more chords that have the same direction and do not intersect inside the circle.
## Relationship Between Chord and Distance to Center
There is an interesting relationship between the length of a chord and its distance from the center of the circle. The closer the chord is to the center of the circle, the longer the chord. This relationship can be expressed mathematically.
If $$d$$ is the distance from the center of the circle to the chord, $$r$$ is the radius of the circle, and $$l$$ is the length of the chord, then:
Visible text: If is the distance from the center of the circle to the chord, is the radius of the circle, and is the length of the chord, then:
```math
l = 2\sqrt{r^2 - d^2}
```
This formula shows that when $$d = 0$$ (chord passes through the center), then $$l = 2r$$ which is the diameter.
Visible text: This formula shows that when (chord passes through the center), then which is the diameter.
Component: LineEquation
Props:
- title: Relationship Between Chord Length and Distance to Center
- description: Chords with different distances from the center of the circle.
- cameraPosition: [0, 0, 8]
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- data: [
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points: createCircleOutlinePoints(3),
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smooth: true,
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labels: [{ text: "d = 2", at: 0, offset: [0, 0.5, 0] }]
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## Properties of Equal Length Chords
Chords that have the same length in the same circle have special properties. They have the same distance from the center of the circle. Conversely, chords that have the same distance from the center of the circle will have the same length.
This property is very useful in solving various circle geometry problems. If two chords have the same length, then:
1. The distance of both chords to the center of the circle is the same
2. The arcs formed by both chords have the same length
3. The central angles facing both chords have the same measure
## Intersecting Chords Theorem
When two chords intersect inside a circle, there is a special relationship between the segments formed. If chords $$AB$$ and $$CD$$ intersect at point $$P$$, then:
Visible text: When two chords intersect inside a circle, there is a special relationship between the segments formed. If chords and intersect at point , then:
```math
AP \times PB = CP \times PD
```
This theorem is known as the intersecting chords theorem and is very useful in solving various geometry problems.
## Practice Problems
1. A circle has a radius of $$13 \text{ cm}$$. If the distance of a chord to the center of the circle is $$5 \text{ cm}$$, determine the length of the chord.
2. In a circle with radius $$10 \text{ cm}$$, there is a chord with length $$16 \text{ cm}$$. Determine the distance of the chord to the center of the circle.
3. Two chords $$AB$$ and $$CD$$ in the same circle have lengths of $$24 \text{ cm}$$ and $$18 \text{ cm}$$ respectively. If the radius of the circle is $$15 \text{ cm}$$, determine the difference in distance between the two chords to the center of the circle.
4. Chords $$PQ$$ and $$RS$$ intersect at point $$T$$ in a circle. If $$PT = 6 \text{ cm}$$, $$TQ = 8 \text{ cm}$$, and $$RT = 4 \text{ cm}$$, determine the length of $$TS$$.
5. A circle has a diameter of $$26 \text{ cm}$$. Determine the length of a chord that is $$12 \text{ cm}$$ away from the center of the circle.
Visible text: 1. A circle has a radius of . If the distance of a chord to the center of the circle is , determine the length of the chord.
2. In a circle with radius , there is a chord with length . Determine the distance of the chord to the center of the circle.
3. Two chords and in the same circle have lengths of and respectively. If the radius of the circle is , determine the difference in distance between the two chords to the center of the circle.
4. Chords and intersect at point in a circle. If , , and , determine the length of .
5. A circle has a diameter of . Determine the length of a chord that is away from the center of the circle.
### Answer Key
1. **Answer: $$24 \text{ cm}$$**
Given: $$r = 13 \text{ cm}$$, $$d = 5 \text{ cm}$$
Using the chord length formula:
```math
l = 2\sqrt{r^2 - d^2}
```
```math
l = 2\sqrt{13^2 - 5^2}
```
```math
l = 2\sqrt{169 - 25}
```
```math
l = 2\sqrt{144} = 2 \times 12 = 24 \text{ cm}
```
2. **Answer: $$6 \text{ cm}$$**
Given: $$r = 10 \text{ cm}$$, $$l = 16 \text{ cm}$$
Using the chord length formula:
```math
16 = 2\sqrt{10^2 - d^2}
```
```math
8 = \sqrt{100 - d^2}
```
```math
64 = 100 - d^2
```
```math
d^2 = 100 - 64 = 36
```
```math
d = 6 \text{ cm}
```
3. **Answer: $$3 \text{ cm}$$**
Given: $$r = 15 \text{ cm}$$, $$l_1 = 24 \text{ cm}$$, $$l_2 = 18 \text{ cm}$$
Finding the distance of the first chord:
```math
24 = 2\sqrt{15^2 - d_1^2}
```
```math
12 = \sqrt{225 - d_1^2}
```
```math
144 = 225 - d_1^2
```
```math
d_1^2 = 81, \text{ so} d_1 = 9 \text{ cm}
```
Finding the distance of the second chord:
```math
18 = 2\sqrt{15^2 - d_2^2}
```
```math
9 = \sqrt{225 - d_2^2}
```
```math
81 = 225 - d_2^2
```
```math
d_2^2 = 144, \text{ so} d_2 = 12 \text{ cm}
```
Difference in distance: $$d_2 - d_1 = 12 - 9 = 3 \text{ cm}$$
4. **Answer: $$12 \text{ cm}$$**
Given: PT = $$6 \text{ cm}$$, TQ = $$8 \text{ cm}$$, RT = $$4 \text{ cm}$$
Using the intersecting chords theorem:
```math
PT \times TQ = RT \times TS
```
```math
6 \times 8 = 4 \times TS
```
```math
48 = 4 \times TS
```
```math
TS = 12 \text{ cm}
```
5. **Answer: $$10 \text{ cm}$$**
Given: diameter = $$26 \text{ cm}$$, so $$r = 13 \text{ cm}$$, $$d = 12 \text{ cm}$$
Using the chord length formula:
```math
l = 2\sqrt{r^2 - d^2}
```
```math
l = 2\sqrt{13^2 - 12^2}
```
```math
l = 2\sqrt{169 - 144}
```
```math
l = 2\sqrt{25} = 2 \times 5 = 10 \text{ cm}
```
Visible text: 1. **Answer: **
Given: ,
Using the chord length formula:
2. **Answer: **
Given: ,
Using the chord length formula:
3. **Answer: **
Given: , ,
Finding the distance of the first chord:
Finding the distance of the second chord:
Difference in distance:
4. **Answer: **
Given: PT = , TQ = , RT =
Using the intersecting chords theorem:
5. **Answer: **
Given: diameter = , so ,
Using the chord length formula: