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Learn · Mathematics · Grade 11 · Geometry

The circle: angles, chords and tangents

Central and inscribed angles, the tangent–radius right angle, chord and tangent length relations, arc length and sector area.

High school Lesson 30 of 31 Interactive figure Practice

First, the rules

Angles, chords and tangents

Three families of facts: angles standing on an arc, lengths of crossing chords and tangents, and the arc and sector formulas. Each comes with a picture — draw the radii and the facts fall out.

★ Inscribed angle
Example 1
inscribed = 12 · central angle on the same arc
All inscribed angles on the same arc are equal.
★ Angle in a semicircle
Example 1
an angle standing on a diameter is 90°
Cyclic quadrilateral
Example 1
opposite angles add to 180°
★ Tangent and radius
Example 2
tangent ⊥ radius at the point of contact
two tangents from one point are equal: |PA| = |PB|
★ Intersecting chords
Example 2
|PA| · |PB| = |PC| · |PD|
Tangent–secant
Example 2
|PT|² = |PA| · |PB| — T the point of contact, A and B where the secant meets the circle.
★ Arc length and sector area
Example 3
arc = θ360° · 2πr, sector = θ360° · πr²
In radians: arc = rθ, sector = ½ r²θ.
Chord from the centre
Example 3
a perpendicular from the centre bisects the chord — half-chord, distance and radius form a right triangle.

See it

Inscribed angle vs central angle

Drag the three points. However P moves along its arc, angle APB stays half of angle AOB.

Step by step

Worked examples

1

Angles on a circle

inscribed · semicircle

A, B, C, D lie on a circle with centre O. Angle AOB = 110°, and AC is a diameter. Find (a) the inscribed angle ADB on the major arc, (b) angle ABC.

110°55°OABD

Ask yourself: Inscribed angle = half the central angle on the same arc. A diameter gives a right angle.

Show the solution
  1. (a) D sees arc AB with central angle 110°: ∠ADB = 110° / 2 = 55°.
  2. (b) B stands on the diameter AC, so ∠ABC = 90°.
  3. Check: the reflex angle AOB is 250°, so an inscribed angle on the minor arc (the other side) would be 125°; 55° + 125° = 180°, as a cyclic quadrilateral demands.

Answer (a) 55° (b) 90°

2

Chords and a tangent

lengths

(a) Chords AB and CD cross at P with |PA| = 4, |PB| = 9, |PC| = 6. Find |PD|. (b) From a point P, 13 from the centre of a circle of radius 5, a tangent touches the circle at T. Find |PT|.

Ask yourself: (a) Product of the parts of one chord equals that of the other. (b) The radius to T is perpendicular to the tangent.

Show the solution
  1. (a) 4 · 9 = 6 · |PD| ⇒ |PD| = 6.
  2. (b) Triangle OTP is right-angled at T: |PT| = √13² − 5² = √144 = 12 (a 5-12-13 triple).

Answer (a) 6 (b) 12

3

Arc, sector and a chord

measurement

A circle has radius 6. (a) Find the arc length and sector area for a central angle of 120°. (b) A chord is 8 long; how far is it from the centre?

Ask yourself: (a) 120° is one third of the turn. (b) Drop the perpendicular from the centre: it bisects the chord.

Show the solution
  1. (a) arc 120360 · 2π · 6 = 4π, sector 120360 · π · 36 = 12π.
  2. (b) Half-chord 4, radius 6: distance √36 − 16 = √20 = 2√5 ≈ 4.47.

Answer (a) 4π and 12π (b) 2√5

Practice: circle geometry

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Inscribed and central angles, angles in a semicircle, tangent lengths, crossing chords, arcs and sectors — type 4π or a decimal. Your best score is kept in this browser only.

About this lesson

The circle: central and inscribed angles on the same arc, the angle in a semicircle, the right angle between a radius and a tangent, equal tangents from a point, the intersecting chords and tangent–secant relations, arc length and sector area. Drag the points on the circle, then practise.

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