Length of tangent
Length of Tangent
The length of a tangent to a circle is the distance from the point of tangency to the point outside the circle from which the tangent is drawn. A tangent to a circle is a line that touches the circle at exactly one point, and this point is known as the point of tangency.
Properties of Tangents
Before we delve into the length of a tangent, let's review some important properties of tangents to a circle:
- A tangent to a circle is perpendicular to the radius at the point of tangency.
- Tangents drawn from the same external point to a circle are equal in length.
- The angle between a tangent and a chord drawn from the point of tangency is equal to the angle in the alternate segment.
Length of Tangent Formula
If a circle has a radius ( r ) and the distance from the center of the circle to the external point ( P ) is ( d ), then the length of the tangent ( l ) from the point ( P ) to the circle can be found using the Pythagorean theorem.
The formula for the length of the tangent is:
[ l = \sqrt{d^2 - r^2} ]
where:
- ( l ) is the length of the tangent,
- ( d ) is the distance from the center of the circle to the external point ( P ),
- ( r ) is the radius of the circle.
Table of Differences and Important Points
| Property | Tangent | Chord |
|---|---|---|
| Definition | A line that touches the circle at exactly one point. | A line segment with both endpoints on the circle. |
| Length | The length can be calculated using ( l = \sqrt{d^2 - r^2} ). | The length varies and can be calculated using different formulas depending on the given information. |
| Relation to Radius | Perpendicular to the radius at the point of tangency. | Bisected by a radius if the chord is a diameter. |
| Number of Points with Circle | Intersects the circle at one point. | Intersects the circle at two points. |
Examples
Example 1: Finding the Length of a Tangent
Suppose a circle has a radius of 5 units, and there is a point ( P ) that is 13 units away from the center of the circle. To find the length of the tangent from point ( P ) to the circle, we use the formula:
[ l = \sqrt{d^2 - r^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 ]
So, the length of the tangent is 12 units.
Example 2: Tangents from the Same External Point
Let's say we have two tangents ( PA ) and ( PB ) drawn from the same external point ( P ) to a circle with center ( O ). If ( OP ) is 10 units and the radius of the circle is 6 units, then the lengths of both tangents ( PA ) and ( PB ) are equal and can be calculated as:
[ l = \sqrt{d^2 - r^2} = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8 ]
So, both ( PA ) and ( PB ) are 8 units long.
Example 3: Angle Between Tangent and Chord
If a tangent ( PT ) touches a circle at ( T ), and a chord ( TC ) is drawn from the point of tangency, then the angle ( \angle PTC ) is equal to the angle in the alternate segment, which is the angle subtended by the chord ( TC ) at the opposite side of the circle.
For instance, if ( \angle TAC ) (where ( A ) is the point on the circle opposite to ( T )) is 45 degrees, then ( \angle PTC ) will also be 45 degrees.
Understanding the length of a tangent and its properties is crucial for solving problems related to circles in geometry. The above examples illustrate how to apply the formula and properties to calculate the length of tangents and understand their relationship with other elements of a circle.