Tangents and Intersecting Chords

Vardaan Learning Institute • ICSE Class 10 Mathematics • Chapter 18

1. Fundamental Definitions

Secant: A line that intersects a circle in two distinct points.

Tangent: A line in the plane of a circle that touches the circle at exactly one point. This point is called the point of contact.

2. Core Tangent Theorems

Theorem 1: Radius-Tangent Perpendicularity

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

$$\text{If } PT \text{ is a tangent at } T \text{ to circle } C(O, r), \text{ then } OT \perp PT$$

Theorem 2: Tangents from an External Point

If two tangents $PA$ and $PB$ are drawn to a circle from an external point $P$:

The lengths of the tangents are equal: $PA = PB$
They subtend equal angles at the center: $\angle AOP = \angle BOP$
They are equally inclined to the line joining the point to the center: $\angle APO = \angle BPO$

Theorem 3: Alternate Segment Theorem

If a line touches a circle and from the point of contact a chord is drawn, the angles which the chord makes with the tangent line are equal to the angles subtended by the chord in the corresponding alternate segments.

$$\angle BAT = \angle BCA \quad \text{and} \quad \angle BAS = \angle BDA$$

3. Intersecting Chords Theorems

Theorem 4: Internal Intersection of Chords

If two chords $AB$ and $CD$ intersect inside a circle at point $P$, then the rectangles contained by their segments are equal in area:

$$PA \times PB = PC \times PD$$

Theorem 5: External Intersection of Chords (Secants)

If two chords $AB$ and $CD$ intersect outside a circle at point $P$, then:

$$PA \times PB = PC \times PD$$

Theorem 6: Tangent-Secant Theorem

If a tangent $PT$ and a secant $PAB$ are drawn from an external point $P$ to a circle, then the product of the secant segments equals the square of the tangent segment:

$$PT^2 = PA \times PB$$

4. Common Tangents to Two Circles

Type of Tangent	Formula ($d = \text{distance between centers}$, $R, r = \text{radii}$)
Direct Common Tangent (DCT)	$$\text{Length} = \sqrt{d^2 - (R - r)^2}$$
Transverse Common Tangent (TCT)	$$\text{Length} = \sqrt{d^2 - (R + r)^2}$$

5. Solved Board Examples

Problem 1: In a circle with center $O$, a tangent $PT$ of length $12\text{ cm}$ is drawn from an external point $P$. If $OP = 13\text{ cm}$, find the radius of the circle.

Solution:

Since radius is perpendicular to the tangent at the point of contact, $\triangle OTP$ is a right-angled triangle with $\angle OTP = 90^\circ$.

By Pythagoras Theorem: $OP^2 = OT^2 + PT^2$

$$13^2 = OT^2 + 12^2 \implies 169 = OT^2 + 144 \implies OT^2 = 25 \implies OT = 5\text{ cm}$$

Therefore, radius of the circle = $5\text{ cm}$.

Problem 2: A secant $PAB$ intersects a circle at $A$ and $B$ such that $PA = 4\text{ cm}$ and $AB = 5\text{ cm}$. Find the length of the tangent $PT$ drawn from $P$ to the circle.

Solution:

Length $PB = PA + AB = 4 + 5 = 9\text{ cm}$.

By Tangent-Secant Theorem: $PT^2 = PA \times PB$

$$PT^2 = 4 \times 9 = 36 \implies PT = 6\text{ cm}$$

Exam Practice Questions (ICSE PYQs)

PYQ 2023

BOARD Two circles touch externally at point $P$. A direct common tangent touches the circles at $A$ and $B$. Prove that $\angle APB = 90^\circ$.

PYQ 2020

BOARD Chords $AB$ and $CD$ of a circle intersect internally at $P$. If $PA = 6\text{ cm}$, $PB = 4\text{ cm}$, and $PC = 3\text{ cm}$, calculate the length of $PD$.