Circles

Learning Objectives

• Understand the concept of a tangent to a circle as a line that intersects the circle at exactly one point and is perpendicular to the radius at the point of contact.
• Analyze the relationship between secants and tangents, recognizing a tangent as a special case of a secant where the endpoints of the chord coincide.
• Calculate the number of tangents that can be drawn from a point relative to a circle: none from inside, one from on the circle, and two from outside.
• Prove that the lengths of tangents drawn from an external point to a circle are equal.
• Demonstrate the perpendicularity of a tangent to the radius at the point of contact.
• Apply the concept of chord bisected by the radius in two concentric circles, where the chord of the larger circle touching the smaller circle is bisected at the point of contact.

Chapter Notes

Tangent to a Circle

• Definition: A tangent to a circle is a line that intersects the circle at exactly one point.
• Perpendicularity: The tangent is perpendicular to the radius at the point of contact.

Secant and Tangent Relationship

• A tangent is a special case of a secant where the two endpoints of its corresponding chord coincide.

Number of Tangents from a Point

• Inside a Circle: No tangents can be drawn.
• On the Circle: Exactly one tangent can be drawn.
• Outside the Circle: Exactly two tangents can be drawn.

Equal Lengths of Tangents

• The lengths of tangents drawn from an external point to a circle are equal.

Perpendicularity of Tangent and Radius

• Theorem: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
  • Proof: Given a circle with center $O$ and a tangent $XY$ at point $P$, $OP$ is perpendicular to $XY$ because $OP$ is the shortest distance from $O$ to $XY$.

Chord Bisected by Radius

• In two concentric circles, the chord of the larger circle that touches the smaller circle is bisected at the point of contact.
  • Proof: Given two concentric circles with center $O$, and a chord $AB$ of the larger circle touching the smaller circle at $P$, $OP$ is perpendicular to $AB$ and thus bisects it, i.e., $AP = BP$.

Example Problems

• Example 1: Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
  • Solution: Join $OP$. Since $AB$ is a tangent to the smaller circle at $P$ and $OP$ is its radius, $OP \perp AB$. Thus, $AP = BP$.
• Example 2: Two tangents $TP$ and $TQ$ are drawn to a circle with center $O$ from an external point $T$. Prove that $\angle PTQ = 2 \angle OPQ$.
  • Solution: Since $TP = TQ$, $\triangle TPQ$ is isosceles. Therefore, $\angle TPQ = \angle TQP = \frac{1}{2}(180^\circ - \theta) = 90^\circ - \frac{1}{2}\theta$. Also, $\angle OPT = 90^\circ$. Thus, $\angle OPQ = 90^\circ - (90^\circ - \frac{1}{2}\theta) = \frac{1}{2}\theta$, giving $\angle PTQ = 2 \angle OPQ$.
• Example 3: $PQ$ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at $P$ and $Q$ intersect at a point $T$. Find the length $TP$.
  • Solution: Join $OT$. Let it intersect $PQ$ at $R$. Then $\triangle TPQ$ is isosceles and $OT$ bisects $PQ$, giving $PR = RQ = 4$ cm. Also, $OR = \sqrt{5^2 - 4^2} = 3$ cm. Using similarity, $\frac{TP}{5} = \frac{4}{3}$, so $TP = \frac{20}{3}$ cm.