Maths · Suggested Grade 10
Class 10 Circle Tangents: Theorems, Proofs & Practice
Two tangent segments are equal when they start at the same external point and touch the same circle. The shared starting point is part of the theorem, not a detail you can leave out.
Original explanations and practice by BananaLearn · Published 31 October 2026
On a small screen, swipe across the illustration to read every label.
Name the objects before using a theorem
A tangent to a circle meets it at exactly one point, the point of contact. A secant cuts the circle at two points. A radius joins the centre to a point on the circle. The radius to a tangent’s contact point is perpendicular to that tangent.
Let the centre be O, an external point P and a point of contact A. Then OA is perpendicular to PA, so triangle OAP is right-angled at A. The point A must be the contact point; you cannot choose a different point on the circle and assume its radius is also perpendicular to the same tangent.
Why a tangent is perpendicular to the radius
The shortest distance from the centre O to a line is along the perpendicular. If a line through a circle point A were not perpendicular to OA, its perpendicular distance from O would be smaller than the radius. Such a line would pass through the circle rather than just touch it at A.
This geometric argument explains the condition behind the right-angle mark. In a school proof, you can cite the radius–tangent theorem once you have identified the contact point. Do not mark a right angle solely because a diagram appears to show one. State the theorem and match it to the labelled points.
Prove equal tangents with RHS congruence
Draw tangents PA and PB from one external point P to the same circle. Join OA, OB and OP. Angles OAP and OBP are 90°. OA = OB because both are radii, and OP is the common hypotenuse of the two right triangles. By RHS congruence, triangles OAP and OBP are congruent. Therefore PA = PB.
Write the equalities in the order that supports the criterion. The shared OP is not another radius; it extends to the external point. The proof also gives equal angles at P, so OP bisects angle APB. It does not say that tangent segments from different external points have equal lengths.
Worked example: calculate a tangent length
A circle has radius 5 cm and OP = 13 cm. Since OA is perpendicular to PA, apply Pythagoras: PA² = OP² − OA² = 169 − 25 = 144. The segment length PA is 12 cm. The second tangent from the same P also has length 12 cm.
The hypotenuse is OP because it lies opposite the right angle. Adding 169 and 25 would treat the wrong segment as the hypotenuse. A length is non-negative, so use the positive square root. If OP were smaller than the radius, P would not be an external point and this configuration would be impossible.
Worked example: turn equal lengths into an equation
Suppose PA = 3x + 1 and PB = 5x − 9, both tangent segments from P. Set 3x + 1 = 5x − 9. Then 2x = 10, so x = 5. Substitution gives both lengths 16 in the stated length unit.
Check the actual lengths, not only x. An algebraic solution producing a negative segment would not describe the stated geometry. If the expressions belonged to tangents from different starting points, the equality would not be justified without more information.
Worked example: combine central and tangent angles
In quadrilateral OAPB, the angles at A and B are each 90°. The quadrilateral sum is 360°, so angle APB plus the minor central angle AOB is 180°. If APB = 68°, the minor angle AOB is 112°. Since OP bisects APB, angle APO is 34°.
Distinguish the minor central angle from the reflex angle. The reflex angle in this example is 360° − 112° = 248°. A question that asks only for AOB normally needs the labelled diagram or minor/reflex wording to settle which angle is intended. State the version you are calculating.
What a complete proof answer should contain
Name the construction, state the two right angles, identify the equal radii and common hypotenuse, cite RHS, then draw the required conclusion from corresponding parts. A list of true facts without the congruence step does not explain why the tangent lengths follow.
For revision, cover the proof and reconstruct it from the picture. Then change the letters so the learner has to identify the relationships rather than recite a sequence. A parent can ask where each equality comes from. Use the connected Pythagoras and congruence guides when a missing prerequisite makes the proof difficult.
A tangent-proof checklist
Copy the record into a notebook or use it to organise your own answer. Blank spaces are for your work.
| Statement | Reason | Your labelled points |
|---|---|---|
| Two right angles | Radius perpendicular to tangent at contact | ___ |
| Equal radius legs | Radii of the same circle | ___ |
| Common hypotenuse | The same centre-to-external-point segment | ___ |
| Congruence then equal tangents | RHS and corresponding parts | ___ |
10 questions with explained answers
Try the questions before opening the answers. Explain the reason, then try a changed example to check your understanding.
What angle does a radius make with the tangent at its contact point?
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90°
Radius–tangent perpendicularity applies at that contact point.PA and PB are tangents from P. If PA = 9 cm, find PB.
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9 cm
They start at the same external point and touch the same circle.Radius 8 cm, OP = 17 cm. Find PA.
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15 cm
√(17² − 8²) = √225.PA = x + 7 and PB = 3x − 5. Find x.
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6
Equate the two tangent lengths; both then equal 13.Angle APB is 80°. Find the minor angle AOB.
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100°
The two angles sum to 180°.In that configuration find angle APO.
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40°
OP bisects the angle between the tangents.Which congruence criterion proves equal tangents?
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RHS
Right angles, common hypotenuse OP and equal radius-leg.Can tangent lengths from different external points be assumed equal?
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No
The theorem requires the same external point.Can a point 4 cm from the centre of a radius-5 cm circle have real tangent segments?
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No
The point is inside the circle.If the minor central angle AOB is 120°, find the reflex angle and angle APB.
Show answer and explanation
240° and 60°
Reflex = 360° − 120°; tangent angle = 180° − 120°.
Keep the ideas connected
- Review the right-triangle calculation
- Build a congruence proof
- Review radius and circle measurements
- Explore Class 10 Maths support
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