Worked example 1
Question: A circle has radius 5 cm. An external point is 13 cm from its centre. Find the tangent length.
Solution: Radius and tangent are perpendicular. PT=√(OP²−OT²)=√(13²−5²)=√144=12 cm.
What this chapter covers
Apply the tangent–radius theorem and equal tangent lengths from an external point to solve geometric and numerical problems.
Tangent at a point
A tangent touches a circle at exactly one point. The radius drawn to the point of contact is perpendicular to the tangent. This creates a right triangle with the centre, contact point and external point.
Equal tangents
Tangents drawn from the same external point to a circle have equal lengths. Combine this result with angle bisectors or perimeter information in proof problems.
Key formulae: If PT is tangent at T to a circle with centre O, then OT⊥PT and PT²=OP²−r². For tangents PA and PB from P: PA=PB.
Worked examples: CBSE question patterns
These are original practice questions modeled on common board-exam patterns; they are not reproduced past-paper questions.
Worked example 2
Question: Tangents PA and PB are drawn from P. If PA=3x−2 and PB=2x+5, find x and the tangent length.
Solution: PA=PB: 3x−2=2x+5, so x=7. Each tangent is 19 units.
Practical activity / investigation
Draw a circle and choose a point outside it. Construct two tangents with a ruler and compass; measure their lengths and the radius at each contact. Check that the radius is perpendicular to each tangent and both tangent lengths are equal.
Record: Write the aim, materials, steps, observations and conclusion in your notebook. Follow teacher supervision for any classroom activity.
Quick practice with answers
Try each question before opening its solution.
Practice 1: A tangent is 24 cm and the radius is 7 cm. Find the centre-to-external-point distance.
OP=√(24²+7²)=25 cm.
Practice 2: If two tangents from P are 2y+1 and 13, find y.
2y+1=13, so y=6.
Practice 3: State the angle between a radius and tangent at contact.
90°.
Revision checklist
- Can I explain the key idea without memorising a worked solution?
- Have I selected the right formula and written units where needed?
- Can I show each algebraic or geometric step clearly?
- Can I check my answer by substitution, estimation or a diagram?
Aligned to the Class X Mathematics curriculum for 2026–27 and the NCERT Mathematics textbook. Always confirm any school-specific internal assessment with your teacher.