Circles • Topic 2 of 4

Tangents & Secants

A tangent touches the circle at exactly one point and is perpendicular to the radius drawn to that point — this 90° is the single most-used fact in tangent problems, because it instantly creates a right triangle with the radius and the line from the centre to the external point. From an external point P two tangents can be drawn, and they are equal in length: PT = √(OP² − r²). That equal-tangent property powers a whole family of CAT questions on incircles, where the tangent lengths from each vertex of a triangle to the incircle satisfy x + y, y + z, z + x as the three sides. A secant cuts the circle at two points. When a tangent PT and a secant PAB are drawn from the same external point, PT² = PA × PB — the tangent–secant case of the power of a point. Recognising whether a line is tangent or secant decides which relation to use.

✅ Solved examples

1. The radius of a circle is 7 cm and an external point P is 25 cm from the centre. Find the length of the tangent from P.
PT = √(OP² − r²) = √(625 − 49) = √576 = 24 cm.
2. Two tangents from a point P touch a circle of radius 6 cm, and the tangent length is 8 cm. Find OP.
OP = √(r² + PT²) = √(36 + 64) = √100 = 10 cm.
3. A circle is inscribed in a right triangle with legs 6 cm and 8 cm (hypotenuse 10 cm). Find the inradius using tangent lengths.
r = (a + b − c)/2 for a right triangle = (6 + 8 − 10)/2 = 2 cm.
4. From point P, a tangent of length 12 cm and a secant are drawn. The secant’s near intersection is 8 cm from P. Find the far intersection distance from P.
PT² = PA × PB ⇒ 144 = 8 × PB ⇒ PB = 18 cm from P.

✏️ Practice — try these, take hints as needed

1. Radius 9 cm, external point 41 cm from centre. Tangent length?
PT = √(OP² − r²).
√(1681 − 81).
√1600.
40 cm
2. Tangent length from P is 24 cm to a circle of radius 7 cm. Distance OP?
OP² = r² + PT².
49 + 576.
√625.
25 cm
3. A circle inscribed in a triangle touches its sides; the tangent lengths from two vertices are 5 and 7, and the third side is 9. Find the triangle’s perimeter.
Third vertex tangents: let it be z; sides are 5+7, 7+z, 5+z.
One side = 9 ⇒ 5+? no — use 7+z or 5+z = 9 with consistency.
Sides: 12, (7+z), (5+z); set 7+z and 5+z from the figure, z = 2 ⇒ sides 12, 9, 7; perimeter = 2(5+7+2).
28 cm
4. From P, a tangent is 15 cm; a secant cuts the circle at A and B with PA = 9 cm. Find PB.
PT² = PA × PB.
225 = 9 × PB.
Divide.
25 cm
5. Two concentric circles have radii 13 cm and 5 cm. A chord of the larger circle is tangent to the smaller. Find the chord length.
Tangent distance = inner radius = 5.
Half-chord = √(13² − 5²).
√144 = 12, double.
24 cm

📝 Topic test — 8 questions

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