A tangent to a circle is a straight line that touches the circle at exactly one point. This point is known as the point of tangency. The following are key concepts related to tangents:

Definition of a Tangent

A line that intersects a circle at exactly one point is called a tangent to the circle. Mathematically, if a line \( l \) touches a circle at point \( P \), then \( l \) is a tangent to the circle.

Properties of Tangents

• A tangent to a circle is perpendicular to the radius drawn to the point of tangency. If \( O \) is the center of the circle and \( P \) is the point of tangency, then:

$$ OP \perp l $$

• If two tangents are drawn from an external point \( A \) to a circle, then the lengths of the tangents from that point to the points of tangency are equal. If \( B \) and \( C \) are the points of tangency, then:

$$ AB = AC $$

Finding the Length of a Tangent

To find the length of a tangent drawn from a point \( A \) outside a circle with center \( O \) and radius \( r \), we can use the following formula:

Let \( d \) be the distance from point \( A \) to the center \( O \). The length of the tangent \( AT \) from point \( A \) to the point of tangency \( T \) is given by:

$$ AT = \sqrt{d^2 - r^2} $$

Construction of Tangents

To construct a tangent from a point outside the circle:

1. Draw a line segment from the external point \( A \) to the center \( O \) of the circle.
2. Locate the midpoint \( M \) of segment \( AO \).
3. Draw a circle with center \( M \) and radius \( r = MA \). This circle will intersect the given circle at two points.
4. Connect these intersection points to \( A \) to form the tangents.

Example Problems

1. Find the length of the tangent from point \( A(6, 8) \) to the circle with center \( O(2, 4) \) and radius \( r = 3 \).

Solution:

Calculate \( d \):

$$ d = \sqrt{(6 - 2)^2 + (8 - 4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} $$

Then, use the tangent length formula:

$$ AT = \sqrt{d^2 - r^2} = \sqrt{(4\sqrt{2})^2 - 3^2} = \sqrt{32 - 9} = \sqrt{23} $$

2. Prove that the tangent at any point on a circle is perpendicular to the radius at that point.

Given a circle with center \( O \) and a tangent line at point \( P \), we need to show that:

$$ OP \perp l $$

Using the Pythagorean theorem, it can be shown that if a line \( l \) intersects the circle at \( P \), the distance from \( O \) to the line is equal to the radius \( r \), confirming that the angle is \( 90^\circ \).

Applications of Tangents

Tangents are used in various real-world applications, including engineering, architecture, and computer graphics. Understanding tangents is essential for solving complex geometric problems and analyzing curves.