Practice Maths

L23 — Chords and Tangents

Key Terms

Perpendicular from centre
The perpendicular from the centre to a chord always bisects the chord (cuts it in half).
Intersecting chords
If chords AC and BD cross at P inside the circle: AP × PC = BP × PD.
External point
A point outside the circle from which tangents or secants can be drawn.
Equal tangents
Two tangent segments drawn from the same external point are always equal in length.
Secant
A line that intersects a circle at two points. A chord is the segment of a secant inside the circle.
Tangent–secant rule
From external point E: ET² = EA × EB, where T is the tangent point and A, B lie on the secant.

Chord Properties

PropertyStatement
Perpendicular bisector The perpendicular from the centre to a chord bisects the chord.
Equal chords Equal chords are equidistant from the centre.
Intersecting chords If chords AC and BD intersect at P inside the circle: AP × PC = BP × PD.
Secant–secant Two secants from external point E: EA × EB = EC × ED.

Tangent Properties

PropertyStatement
Radius ⊥ tangent A radius drawn to the point of tangency is perpendicular to the tangent.
Equal tangents Two tangents from an external point are equal in length.
Tangent–secant From external point E: ET² = EA × EB (T = tangent point, A and B on secant).
□ □ O E T1 T2 t t ET⊂1; = ET⊂2; (equal tangents from E)
Hot Tip: From any external point, both tangent lengths are equal — use this to set up equations with unknowns. To find the tangent length, draw the radius to the tangent point: you get a right triangle, so Pythagoras applies directly.

Perpendicular from Centre to Chord

If OM ⊥ AB (where O is the centre and M is the midpoint of chord AB), then OM bisects AB. Conversely, a line from the centre that bisects a chord is perpendicular to it.

Worked Example 1 — Chord length

A circle has radius 13 cm. A chord is 10 cm from the centre. Find the chord length.

Half-chord² = 13² − 10² = 169 − 100 = 69. Half-chord = √69 ≈ 8.31 cm. Chord = 2×8.31 ≈ 16.61 cm.

Intersecting Chords

Worked Example 2

Chords PQ and RS intersect inside a circle at X. PX=4, XQ=9, RX=6. Find XS.

PX × XQ = RX × XS ⇒ 4 × 9 = 6 × XS ⇒ XS = 36/6 = 6.

Tangent from External Point

Worked Example 3

From external point E, a tangent ET = 12 cm and the distance to the near end of a secant EA = 6 cm. Find EB (far end).

ET² = EA × EB ⇒ 144 = 6 × EB ⇒ EB = 24 cm.

Equal Tangents

From any external point, the two tangent segments to a circle are equal. This creates an isosceles triangle (the two tangent lines and the line joining the two tangent points).

  1. Perpendicular from centre to chord. Fluency

    • (a) Circle radius 10 cm. Distance from centre to chord = 6 cm. Find the chord length.
    • (b) Circle radius 17 cm. Chord length = 16 cm. Find the distance from the centre to the chord.
    • (c) Circle radius 25 cm. Chord length = 14 cm. Find the distance from the centre to the chord.
    • (d) Two equal chords are each 24 cm from the centre of a circle of radius 25 cm. Find their lengths.

  2. Intersecting chords. Fluency

    Chords intersect inside the circle at point X.

    • (a) AX=3, XC=12, BX=4. Find XD.
    • (b) PX=6, XR=8, QX=4. Find XS.
    • (c) Both chords are equal. The first is split 5 and 9 by X. Find how the second is split.
    • (d) AX=x, XC=4, BX=(x+2), XD=3. Find x.

  3. Tangents from an external point. Fluency

    • (a) Two tangents from point P to a circle. One tangent length is 15 cm. Find the other.
    • (b) Tangent from E = 8 cm. Distance from E to the near point of a secant = 4 cm. Find the distance to the far point.
    • (c) External point E, secant EA=5, EB=20. Find the tangent length ET.
    • (d) External point P is 13 cm from the centre of a circle of radius 5 cm. Find the tangent length.

  4. Radius perpendicular to tangent. Fluency

    • (a) A tangent touches a circle (r=8 cm) at T. The external point is P with PT=15 cm. Find the distance OP.
    • (b) O is the centre, OT ⊥ tangent, OT=6 cm, external point P, OP=10 cm. Find PT.
    • (c) A tangent from external point E makes an angle of 30° with line OE. The radius is 5 cm. Find OE and the tangent length.
    • (d) Two tangents from P touch a circle at A and B. Angle APB = 60°. The radius is 7 cm. Find PA and OA.

  5. Chord and tangent from a diagram. Understanding

    The diagram shows a circle with centre O and radius 10 cm. Chord AB has its midpoint at M, and OM = 6 cm. Tangent from external point P touches the circle at T. OP = 26 cm.

    O M A B P T 6 cm 26 cm 10 cm
    • (a) Find the length of the chord AB. Show your working.
    • (b) Find the tangent length PT.
    • (c) Angle OAB: the line OA is a radius. Find angle OAB and hence angle AOM.
    • (d) From external point P, a secant passes through the circle intersecting at X (near) and Y (far). PX = 2 cm, XY = 23 cm. Verify that PT² = PX × PY.

  6. Chord distance and equal chords. Understanding

    • (a) A circle has two parallel chords of length 16 cm and 12 cm. The radius is 10 cm. Find the distance between the chords if they are on the same side of the centre.
    • (b) Same as (a) but the chords are on opposite sides of the centre.
    • (c) A chord of length 24 cm is 5 cm from the centre. Find the radius.
    • (d) Two equal chords in a circle are equidistant from the centre. The chords are 18 cm long and the radius is 15 cm. Find the distance from each chord to the centre.

  7. Tangent angle problems. Understanding

    • (a) From external point P, two tangents touch a circle at A and B. OA=OB=5 cm (radii), OP=13 cm. Find the angle between the two tangents (angle APB).
    • (b) In (a), find angle AOB.
    • (c) A tangent from P and a secant from P (through centre O) meet the circle at T, A (near) and B (far) respectively. PA=4 cm, AB=12 cm. Find PT.
    • (d) The tangent from P has length 24 cm and makes an angle of 28° with OP. Find OP and the radius of the circle.

  8. Intersecting secants from external point. Understanding

    • (a) From external point E, two secants: EA=4, EB=9 and EC=3. Find ED.
    • (b) EA=6, EB=24, EC=8. Find ED.
    • (c) EA=5, EB=20 and EC=ED (the secant is a tangent). Find the tangent length.
    • (d) Explain why the intersecting-chords rule and the secant-secant rule are the same relationship, just applied at different points (inside vs outside the circle).

  9. Combined chord and tangent. Problem Solving

    A circle has centre O and radius r. A chord AB of length 2a is at distance d from the centre (so r² = a² + d²). A tangent is drawn from an external point P that lies on the perpendicular bisector of AB, at distance h from the midpoint M of AB (so PM = h, OM = d, OP = d + h).

    • (a) Write an expression for the tangent length PT in terms of r and OP = d + h.
    • (b) Using r² = a² + d², simplify the tangent length in terms of a, d, h.
    • (c) For r=5 cm, a=4 cm, h=7 cm, find d and then PT numerically.
    • (d) For what value of h is PT equal to the half-chord length a?

  10. Cyclic quadrilateral and tangent. Problem Solving

    ABCD is a cyclic quadrilateral inscribed in a circle with centre O. A tangent to the circle at A makes an angle of 40° with AB.

    • (a) Use the alternate segment theorem to find angle ACB.
    • (b) Find the central angle AOB.
    • (c) If angle ABC = 85°, find angle ADB using the same-segment theorem.
    • (d) Find angle ADC (opposite angle in the cyclic quadrilateral to ABC).
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