Circles: Chords, Tangents, Angles, Cyclic Quadrilateral and Common Tangents with Formulas and Solved Examples for SSC

Circle geometry is a favourite of the SSC CGL and CHSL papers, and the questions follow a fixed set of theorems. This guide on circles for SSC covers chords, the angle theorems, the cyclic quadrilateral, the tangent properties, the common tangents and the intersecting chord and secant–tangent theorems, with all the formulas, solved examples, practice questions and FAQs.

Basic Terms

Properties of Chords

• The perpendicular from the centre to a chord bisects the chord; the line from the centre to the midpoint of a chord is perpendicular to it.
• Equal chords are equidistant from the centre, and chords equidistant from the centre are equal.
• A longer chord is nearer to the centre; the diameter is the longest chord.
• Equal chords subtend equal angles at the centre.
• Distance of a chord from the centre: d = √(r² − (c/2)²), where c is the length of the chord.
• Length of a chord: c = 2r sin(θ/2), where θ is the angle at the centre.

Angle Theorems

Tangent Properties

• A tangent is perpendicular to the radius at the point of contact.
• The two tangents drawn from an external point are equal in length, and they subtend equal angles at the centre; the line joining the centre to the external point bisects the angle between them.
• Length of the tangent from a point at a distance d from the centre: √(d² − r²).
• The angle between two tangents from an external point P and the angle at the centre between the radii to the contact points are supplementary (sum 180°).

Two Circles

Intersecting Chords, Secants and Tangents

• Intersecting chords: if chords AB and CD meet at P inside the circle, PA × PB = PC × PD.
• Secants: if secants from an external point P cut the circle at A, B and C, D, then PA × PB = PC × PD (PB and PD are the whole secants).
• Secant and tangent: PT² = PA × PB, where PT is the tangent and PB is the whole secant.
• Ptolemy’s theorem: for a cyclic quadrilateral, the product of the diagonals = the sum of the products of the opposite sides.

Arc, Sector and Circumference

Solved Examples

Example 1. A chord of length 16 cm is in a circle of radius 10 cm. Find its distance from the centre.
d = √(100 − 64) = 6 cm.

Example 2. Find the length of the tangent from a point 13 cm from the centre of a circle of radius 5 cm.
√(169 − 25) = 12 cm.

Example 3. Two circles of radii 8 cm and 3 cm have their centres 13 cm apart. Find the lengths of the direct and the transverse common tangents.
Direct: √(169 − 25) = 12 cm; transverse: √(169 − 121) = √48 = 6.93 cm.

Example 4. Chords AB and CD intersect at P. If PA = 4, PB = 6 and PC = 3, find PD.
4 × 6 = 3 × PD, so PD = 8.

Example 5. If a central angle on an arc is 80°, find the angle at the circumference on the same arc.
40°.

Example 6. In a cyclic quadrilateral ABCD, angle A = 70°. Find angle C.
180 − 70 = 110°.

Quick Tips

• Learn the number of common tangents for each position of two circles.
• Learn the two formulas of the common tangents.
• Use the central angle theorem: angle at the centre = 2 × angle on the circumference.
• Remember that the angle in a semicircle is 90°.

Quick Revision Points

• The perpendicular from the centre bisects a chord.
• The angle at the centre is twice the angle at the circumference.
• The opposite angles of a cyclic quadrilateral add up to 180°.
• The tangents from an external point are equal.
• PA × PB = PC × PD for the intersecting chords.

Practice Questions With Answers

1. What is the angle in a semicircle?
   Answer: 90°
2. What is the length of the tangent from a point 17 cm from the centre of a circle of radius 8 cm?
   Answer: 15 cm
3. How many common tangents do two externally touching circles have?
   Answer: 3
4. What is the sum of the opposite angles of a cyclic quadrilateral?
   Answer: 180°
5. What is the length of the direct common tangent if d = 10, r₁ = 7 and r₂ = 1?
   Answer: 8
6. A chord of 24 cm is in a circle of radius 13 cm. Find its distance from the centre.
   Answer: 5 cm
7. If PT is a tangent and PAB a secant with PA = 4 and PB = 9, find PT.
   Answer: 6
8. What is the angle between a tangent and the radius at the point of contact?
   Answer: 90°
9. What is the area of a sector with radius 7 cm and angle 90°?
   Answer: 38.5 sq cm
10. Which theorem relates the tangent-chord angle to the angle in the alternate segment?
    Answer: Alternate segment theorem

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Mathematics notes page, read the previous topic Triangles and the next topic Quadrilaterals and Polygons.

Frequently Asked Questions

What is a chord?

A line segment joining two points on a circle.

What is a tangent?

A line that touches a circle at exactly one point.

What is the angle subtended by a diameter?

90 degrees on the circumference.

What is a cyclic quadrilateral?

A quadrilateral whose four vertices lie on a circle.

What is the central angle theorem?

The angle at the centre is twice the angle at the circumference on the same arc.

How many tangents can be drawn from an external point?

Two, of equal length.

What is the alternate segment theorem?

The angle between a tangent and a chord equals the angle in the alternate segment.

What is the direct common tangent formula?

√(d² − (r₁ − r₂)²).

What is the intersecting chord theorem?

PA × PB = PC × PD.

Is this topic asked in SSC?

Yes. Circle geometry is asked every year in SSC CGL and CHSL.

What is the length of an arc?

(θ/360) × 2πr.

What is Ptolemy’s theorem?

For a cyclic quadrilateral, the product of the diagonals equals the sum of the products of the opposite sides.