Common chord
Understanding the Common Chord in Circles
In the context of geometry, a chord of a circle is a straight line segment whose endpoints both lie on the circle. When two circles intersect, they may share a common chord. This common chord is significant in various geometric constructions and theorems.
Properties of a Chord
Before diving into the common chord, let's review some basic properties of a chord in a single circle:
- A chord divides the circle into two segments: the major segment and the minor segment.
- The perpendicular bisector of a chord passes through the center of the circle.
- The longest chord of a circle is the diameter.
- Chords equidistant from the center of a circle are equal in length.
Common Chord Between Two Circles
When two circles intersect, the line segment that joins the points of intersection is called the common chord. The common chord has several interesting properties and can be used to derive various geometric relationships.
Properties of the Common Chord
- The common chord is perpendicular to the line joining the centers of the two circles.
- The common chord is bisected by the line joining the centers of the two circles.
Formulas Involving Common Chords
If two circles with radii $r_1$ and $r_2$ and distance $d$ between their centers intersect, the length of the common chord ($l$) can be found using the following formula:
$$ l = 2\sqrt{h\left(\frac{r_1 + r_2 - d}{2}\right)} $$
where $h$ is the distance from the center of either circle to the common chord.
Table of Differences and Important Points
| Property | Chord in a Single Circle | Common Chord Between Two Circles |
|---|---|---|
| Definition | A line segment with both endpoints on the circle | A line segment that is the intersection of two circles |
| Bisector | The perpendicular bisector of a chord passes through the circle's center | The line joining the centers of the two circles bisects the common chord |
| Length Calculation | Not directly related to another circle | Depends on the radii of both circles and the distance between their centers |
| Relation to Circle Centers | Chords equidistant from the center are equal | The common chord is perpendicular to the line joining the centers |
Examples to Explain Important Points
Example 1: Perpendicularity of the Common Chord
Consider two circles with centers $O_1$ and $O_2$ that intersect at points $A$ and $B$, forming the common chord $AB$. The line $O_1O_2$ joining the centers of the two circles will always be perpendicular to the common chord $AB$.
Example 2: Length of the Common Chord
Let's say we have two circles with radii $r_1 = 5$ units and $r_2 = 3$ units, and the distance between their centers $d = 6$ units. To find the length of the common chord, we first need to find the distance $h$ from the center of either circle to the common chord.
Using the formula for the length of the common chord:
$$ h = \frac{r_1^2 - (\frac{d - r_2}{2})^2}{d} $$
Substituting the values, we get:
$$ h = \frac{5^2 - (\frac{6 - 3}{2})^2}{6} = \frac{25 - (\frac{3}{2})^2}{6} = \frac{25 - \frac{9}{4}}{6} = \frac{100 - 9}{24} = \frac{91}{24} $$
Now, we can calculate the length of the common chord $l$:
$$ l = 2\sqrt{h\left(\frac{r_1 + r_2 - d}{2}\right)} = 2\sqrt{\frac{91}{24}\left(\frac{5 + 3 - 6}{2}\right)} = 2\sqrt{\frac{91}{24}\left(\frac{2}{2}\right)} = 2\sqrt{\frac{91}{24}} $$
Thus, the length of the common chord is $2\sqrt{\frac{91}{24}}$ units.
Understanding the concept of a common chord is essential for solving problems related to intersecting circles in geometry. It is a fundamental concept that has applications in various fields, including engineering and design.