One-Line Definition
A Circle is the set of all points equidistant ($r$) from a fixed center. A Chord connects any two boundary points, a Secant cuts through the circle, and a Tangent touches the circle at exactly one point, forming a $90^\circ$ angle with the radius.
Circle Symmetry Models
Mastering Circle Geometry relies on 3 structural rules:
- • Chord Bisector: Perpendicular drawn from center to any chord bisects the chord into two equal halves.
- • Inscribed Angle Double Rule: Angle subtended by an arc at the center is TWICE the angle subtended at any point on the circumference ($\angle \text{Center} = 2 \angle \text{Circumference}$).
- • Alternate Segment Theorem: Angle between tangent and chord equals the angle subtended by the chord in the alternate segment.
Circles and Tangents feature in 3-4 questions in CAT Quant & Data Interpretation every year. Click below to explore connected Quant Vault topics:
Where Is This Used in Real Life & Business?
1 Circle Metrics, Arc Length, Sector & Segment Area
For a circle with radius $r$ and central angle $\theta$ (in degrees):
🔪 Area of Minor Segment
2 Intersecting Chords & Tangent-Secant Theorem
When two lines intersect inside or outside a circle, their segment lengths satisfy crucial product theorems:
If chords $AB$ and $CD$ intersect at point $P$ inside the circle:
If secants $PAB$ and $PCD$ intersect at external point $P$:
⚡ Tangent-Secant Theorem
If a tangent from external point $P$ touches the circle at $T$, and a secant from $P$ cuts the circle at $A$ and $B$:
3 Direct & Transverse Common Tangent Lengths
For two non-intersecting circles with radii $R$ and $r$ whose centers are separated by distance $d$:
1. Direct Common Tangent (DCT)
Tangents that do NOT cross the line segment connecting the two centers.
2. Transverse Common Tangent (TCT)
Tangents that CROSS the line segment connecting the two centers.
🌿 Alternate Segment Theorem
The angle between a tangent and a chord drawn through the point of contact is equal to the angle subtended by that chord in the alternate segment of the circle.
4 Cyclic Quadrilaterals & Ptolemy's Theorem
Cyclic Quadrilateral Properties
A quadrilateral whose all 4 vertices lie on a single circle.
- • Opposite Angles: $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$.
- • Exterior Angle: Exterior angle equals opposite interior angle.
Ptolemy's Theorem
For any cyclic quadrilateral $ABCD$ with diagonals $AC$ and $BD$:
Product of diagonals equals the sum of products of opposite sides!
Interactive Circle & Tangent Solver
Enter Circle Radius & Center Distance to compute Area, Arc Length, Sector Area & Tangent Lengths!
Circle Rules & Touch Counts
-
• Number of Common Tangents:
- $d > R + r$ (Separate): 4 tangents (2 DCT, 2 TCT).
- $d = R + r$ (Touch externally): 3 tangents.
- $d < R + r$ (Intersect): 2 tangents (2 DCT).
- $d = R - r$ (Touch internally): 1 tangent. - • Tangent Perpendicularity: Radius to point of contact is ALWAYS $90^\circ$.
Exam Traps to Avoid
- • Mixing $(R-r)$ and $(R+r)$ in Tangents: Direct Common Tangent uses $(R-r)^2$, whereas Transverse Common Tangent uses $(R+r)^2$!
- • Forgetting $PA \cdot PB = PT^2$: $PA$ and $PB$ MUST be measured from external point $P$ to the TWO intersection points $A$ and $B$ on the circle!
📝 Practice Questions (Level 0 to Level 2)
Q1. Find the area of a sector of a circle of radius 14 cm if the central angle is $60^\circ$. (Use $\pi = 22/7$).
VIEW SOLUTION & STEP-BY-STEP PROOF ▼
Step 1: Use Sector Area Formula
$$\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 = \frac{60^\circ}{360^\circ} \times \frac{22}{7} \times 14 \times 14$$
$$\text{Area} = \frac{1}{6} \times \frac{22}{7} \times 196 = \frac{1}{6} \times 616 = 102.67\text{ cm}^2$$
Answer: $102.67\text{ cm}^2$ (or $\frac{308}{3}\text{ cm}^2$).
Q2. From an external point $P$, a tangent $PT$ of length 12 cm is drawn to a circle. A secant $PAB$ passes through the center of the circle. If the external segment $PA = 8$ cm, find the radius of the circle.
VIEW SOLUTION & STEP-BY-STEP PROOF ▼
Step 1: Apply Tangent-Secant Theorem
$$PT^2 = PA \cdot PB \implies 12^2 = 8 \cdot PB \implies 144 = 8 \cdot PB \implies PB = 18\text{ cm}$$
Step 2: Relate $PB$ to Diameter $AB$
$$PB = PA + AB \implies 18 = 8 + 2r \implies 2r = 10 \implies r = 5\text{ cm}$$
Answer: Radius $r = 5\text{ cm}$.
Q3. Two circles of radii 8 cm and 3 cm have their centers separated by 13 cm. Find the length of the Direct Common Tangent (DCT) and the Transverse Common Tangent (TCT).
VIEW SOLUTION & STEP-BY-STEP PROOF ▼
Step 1: Calculate Direct Common Tangent (DCT)
$$\text{DCT} = \sqrt{d^2 - (R - r)^2} = \sqrt{13^2 - (8 - 3)^2} = \sqrt{169 - 25} = \sqrt{144} = 12\text{ cm}$$
Step 2: Calculate Transverse Common Tangent (TCT)
$$\text{TCT} = \sqrt{d^2 - (R + r)^2} = \sqrt{13^2 - (8 + 3)^2} = \sqrt{169 - 121} = \sqrt{48} = 4\sqrt{3} \approx 6.93\text{ cm}$$
Answer: $\text{DCT} = 12\text{ cm}$, $\text{TCT} = 4\sqrt{3}\text{ cm}$.
❓ Frequently Asked Questions
What is the Alternate Segment Theorem? ▼
It states that the angle between a tangent and a chord through the point of contact is equal to the angle subtended by that chord in the alternate segment of the circle.
How many common tangents can two circles have? ▼
If circles do not touch ($d > R+r$), they have 4 common tangents. If touching externally ($d = R+r$), 3. If intersecting ($R-r < d < R+r$), 2. If touching internally ($d = R-r$), 1. If one inside another ($d < R-r$), 0.