Length of a tangent from an external point: Class 10 practice
A tangent is perpendicular to the radius at the point of contact, so the radius, tangent and the line to the centre form a right triangle: tangent² = d² − r². Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Pythagoras' theorem
The idea behind length of a tangent from an external point
A tangent is perpendicular to the radius at the point of contact, so the radius, tangent and the line to the centre form a right triangle: tangent² = d² − r². The two tangents from a point are equal.
Step-by-step method
- Draw the radius to the point of contact (90°).
- Use tangent² = d² − r².
- Use 'equal tangents' for perimeter problems.
Worked example 1 (Basic)
A point is 17 cm from the centre of a circle of radius 8 cm. Find the length of the tangent.
- t² = 17² − 8² = 225
- t = 15 cm
- Answer: 15 cm
Worked example 2 (Standard)
The tangent from P to a circle is 48 cm and P is 50 cm from the centre. Find the radius.
- r² = 50² − 48² = 196
- r = 14 cm
- Answer: 14 cm
Worked example 3 (Challenge)
A triangle's sides touch a circle; the tangent lengths from its vertices are 6, 6 and 7 cm. Find its perimeter.
- Each side = sum of two tangent lengths
- Perimeter = 2(6 + 6 + 7) = 38 cm
- Answer: 38 cm
Common mistakes and misconceptions
- Adding the squares
- Forgetting the 90° angle
Practice questions
- (Basic) A point is 39 cm from the centre of a circle of radius 15 cm. Find the length of the tangent.
- (Basic) A point is 25 cm from the centre of a circle of radius 7 cm. Find the length of the tangent.
- (Standard) The tangent from P to a circle is 15 cm and P is 17 cm from the centre. Find the radius.
- (Standard) The tangent from P to a circle is 24 cm and P is 25 cm from the centre. Find the radius.
- (Challenge) A triangle's sides touch a circle; the tangent lengths from its vertices are 3, 9 and 8 cm. Find its perimeter.
- (Challenge) A triangle's sides touch a circle; the tangent lengths from its vertices are 3, 5 and 3 cm. Find its perimeter.
Answer key with full working
- Q1: t² = 39² − 15² = 1296 → t = 36 cm → Answer: 36 cm
- Q2: t² = 25² − 7² = 576 → t = 24 cm → Answer: 24 cm
- Q3: r² = 17² − 15² = 64 → r = 8 cm → Answer: 8 cm
- Q4: r² = 25² − 24² = 49 → r = 7 cm → Answer: 7 cm
- Q5: Each side = sum of two tangent lengths → Perimeter = 2(3 + 9 + 8) = 40 cm → Answer: 40 cm
- Q6: Each side = sum of two tangent lengths → Perimeter = 2(3 + 5 + 3) = 22 cm → Answer: 22 cm
Syllabus note
This skill is listed under "Circles" for Class 10 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is length of a tangent from an external point in?
"Circles", Class 10 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.