Class 10 Circle Tangent Theorems: Lengths and Angles

Add the radius to the point of contact

If PT is tangent at T and O is the centre, OT is perpendicular to PT. This gives a right triangle OPT when P is outside the circle. The radius must end at the contact point; a radius to another point does not create this right angle.

Find a length with a right triangle

Take OP = 13 cm and radius OT = 5 cm. OP is the hypotenuse because it is opposite the right angle at T. Thus PT² = 13² − 5² = 144, so PT = 12 cm. Adding the two squares instead would make the tangent longer than the hypotenuse.

The arithmetic checks the length, but the geometric reason belongs in your answer: “radius perpendicular to tangent at the point of contact”. Without it, Pythagoras appears as an unsupported step.

Connect the sketch to the calculation

In the sketch, OP joins the centre to the external point and is the hypotenuse; OT is a radius, not a tangent length. Substituting OP = 13 cm and OT = 5 cm into PT² = OP² − OT² gives PT = 12 cm. The drawing is schematic, so do not measure it to infer these numbers.

Use the same external point

If PA and PB are tangents from the same external point P, then PA = PB. The qualification matters: tangents drawn from different external points need not have equal lengths. To justify the theorem, join OA, OB and OP. The two right triangles share hypotenuse OP and have equal radii, giving RHS congruence.

Decide which fact helps

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An angle check

If angle AOB is 116° for two contact radii, the angle APB between tangents is 360° − 90° − 90° − 116° = 64°. A quick sketch should show that the central angle and this tangent angle add to 180°. Label the intended central angle carefully.

Choose what to study next

Sources and scope

This is a focused learning resource, not a complete chapter or a prediction of board questions. Use your current syllabus and the paper-specific marking scheme for assessment requirements.