ICSE Class 10 Maths: Circles – Tangents & Chord Properties

Ever wondered why a straight line just barely kisses a circle at one point? That tiny kiss is a tangent, and it hides some neat tricks that show up in every ICSE exam.

💡 In Simple Words: A tangent touches a circle at exactly one point, while a chord is a straight line that cuts the circle at two points. Both have special rules that let you find unknown lengths or angles without measuring.

What is a Tangent to a Circle?

A tangent is a line that meets a circle at one and only one point. Think of a road that just skims the edge of a round lake without ever going into the water – that point where the road touches the shore is like the point of tangency.

Key Properties of a Tangent

• The radius drawn to the point of tangency is perpendicular (forms a right angle) to the tangent line.
• From an external point, all tangents drawn to the same circle have equal lengths.
• If a tangent meets a chord at its endpoint, the angle formed equals the angle in the alternate segment (the tangent‑chord theorem).

Understanding Chords of a Circle

A chord is any straight line whose ends both lie on the circle. Picture a piece of string stretched across a round pizza – that string is the chord.

Important Chord Theorems

• The perpendicular from the centre to a chord bisects (splits) the chord.
• Equal chords are equally distant from the centre.
• The angle subtended by a chord at the centre is twice the angle subtended on the circumference (the central‑angle theorem).

Quick Comparison: Tangent vs. Chord

Worked Example 1: Length of a Tangent from an External Point

Given: Point P is 13 cm from the centre O of a circle with radius 5 cm. Find the length of the tangent PT.

Solution steps:

1. Draw OP and the tangent PT. Join OT (radius to point of tangency T).
2. Notice that ΔOPT is a right‑angled triangle because OT ⟂ PT.
3. Apply Pythagoras theorem: PT² = OP² – OT².
4. Substitute: PT² = 13² – 5² = 169 – 25 = 144.
5. Take square root: PT = 12 cm.

Worked Example 2: Angle Between a Tangent and a Chord

Given: In circle O, chord AB subtends a 50° angle at the centre. Find the angle between the tangent at A and chord AB.

Solution:

• The angle subtended by AB at the centre is 50°.
• According to the tangent‑chord theorem, the angle between the tangent at A and chord AB equals the angle in the opposite segment, i.e., the angle subtended by AB on the far side of the circle.
• That opposite angle = ½ (180° – 50°) = 65°.
• Therefore, the required angle is 65°.

Bullet‑Point Summary for Quick Revision

• Tangent touches a circle at exactly one point; radius to that point is perpendicular.
• All tangents from the same external point are equal.
• Chord’s midpoint lies on a line drawn from the centre that is perpendicular to the chord.
• Equal chords are equally distant from the centre.
• Central angle = 2 × inscribed angle on the same chord.
• Tangent‑chord angle equals the angle in the alternate segment.

📝 Likely Exam Questions

1. Question: From a point 15 cm away from the centre of a circle of radius 9 cm, two tangents are drawn. Find the length of each tangent.
   Answer: Use PT² = OP² – r² → PT² = 15² – 9² = 225 – 81 = 144 → PT = 12 cm.
2. Question: In a circle, a chord of length 10 cm is 6 cm from the centre. Find the radius of the circle.
   Answer: Let r be radius, then by right‑triangle formed: (r)² = (5)² + (6)² → r² = 25 + 36 = 61 → r = √61 cm.
3. Question: Prove that the angle between a tangent and a chord equals the angle in the alternate segment.
   Answer: Draw the triangle formed by the radius to the point of tangency and use the fact that the radius is perpendicular to the tangent; then apply the cyclic quadrilateral theorem to show the two angles are equal.
4. Question: Two chords AB and CD intersect at point E inside the circle. If AE = 3 cm, EB = 4 cm, CE = 2 cm, find ED.
   Answer: Use intersecting chords theorem: AE·EB = CE·ED → 3·4 = 2·ED → ED = 6 cm.