What is the Type of Tangents to Circles?

Step-by-step Guide: Tangents to Circles

Definition of a Tangent:
A tangent to a circle is a straight line that touches the circle at exactly one point. This point is called the point of tangency. Unlike chords, tangents never cross through the circle; they merely “kiss” its outer edge.

Properties of Tangents:

• Perpendicular to Radius: The radius drawn to the point of tangency is always perpendicular to the tangent. In other words, the angle between the radius and the tangent at the point of tangency is always \(90^\circ\).
• Equal Lengths: If two tangents are drawn from a single external point to a circle, they are congruent.

Tangents and Secants:
When a tangent and a secant (a line that intersects the circle at two points) meet outside a circle, the measure of the angle formed is half the difference of the measures of the intercepted arcs.

Two Tangent Theorem:
If two tangents are drawn to a circle from an external point, then:

• The tangents are congruent.
• The segments joining the center of the circle and the points of tangency form two congruent angles.

Examples

Example 1:
From a point \( P \) outside a circle with center \( O \), two tangents \( PA \) and \( PB\) are drawn to the circle. If \( PA = 5 \text{ cm} \), find the length of \( PB \).

Solution:
Using the property that tangents drawn from a single external point to a circle are congruent, we can say:
\( PA = PB \)
Given \( PA = 5 \text{ cm} \), \( PB \) is also \( 5 \text{ cm} \).

Example 2:
A tangent \( PT \) is drawn to a circle with center \( O \) at point \( T \). If \( OT = 8 \text{ cm} \) and is perpendicular to \( PT \), find the distance from the center \( O \) to the tangent.

Solution:
Since the radius at the point of tangency is perpendicular to the tangent, the distance from the center \( O \) to the tangent \( PT \) is the same as the length of the radius \( OT \).
Given \( OT = 8 \text{ cm} \), the distance from \( O \) to \( PT \) is \( 8 \text{ cm} \).

Practice Questions:

1. If two tangents are drawn from an external point and they meet the circle at \( A \) and \( B \) respectively, and \( AB = 10 \text{ cm} \), what is the length of each tangent segment?
2. A tangent at \( T \) and a secant through points \( A \) and \( B \) intersect outside a circle at \( P \). If arc \( ATB \) measures \( 140^\circ \) and arc \( AB \) measures \( 60^\circ \), what is the measure of \( \angle PTB \)?
3. From an external point \( P \), two tangents are drawn to a circle of radius \( 6 \text{ cm} \). If the distance between \( P \) and the center of the circle is \( 10 \text{ cm} \), find the length of one of the tangent segments.

Answers:

1. \( 5 \text{ cm} \) for each segment (since the two tangents are congruent).
2. \( 40^\circ \) (half the difference of the measures of the intercepted arcs).
3. \( 8 \text{ cm} \) (using the Pythagorean theorem).