Circles: angles, chords and tangents · Year 11
In Circles: angles, chords and tangents (Year 11) you practise 7 skills: Central and inscribed angles · The angle in a semicircle · Chords and their distance from the centre · Cyclic quadrilaterals · Tangents to a circle · A quadrilateral round a circle · The angle between a tangent and a chord. A central angle has its vertex at the centre of the circle. An inscribed angle has its vertex on the circle, and its arms are chords. An inscribed angle is half the central angle standing on the same arc, and inscribed angles standing on the same arc are equal. An inscribed angle standing on a diameter is a right angle. In a quadrilateral inscribed in a circle, each pair of opposite angles adds up to 180°, and the converse holds too.
The perpendicular from the centre of a circle to a chord halves the chord. Half the chord, its distance from the centre and the radius make a right triangle, so Pythagoras’ theorem finds any one of them from the other two. Equal chords are equally far from the centre, and a longer chord is nearer to it. A tangent touches a circle at one point, and it is perpendicular to the radius there. Two tangents to a circle from the same point are equal. So in a quadrilateral drawn round a circle, the sums of opposite sides are equal. The angle between a tangent and a chord from the point of contact equals the inscribed angle standing on the arc inside it. A model proof that an inscribed angle is half the central angle on the same arc, when the centre lies on one arm of the inscribed angle: join the centre to the far end of the other arm. The two radii make an isosceles triangle, so its base angles are equal. The central angle is an exterior angle of that triangle, so it equals the sum of the two base angles, which is twice the inscribed angle. When the centre lies inside the angle or outside it, draw the diameter through the vertex and add or subtract two angles of this kind.
Tip: Mark the centre and the radii: equal radii make isosceles triangles, and a radius to a point of contact makes a right angle.
What you learn
Central and inscribed angles
For example: In a circle with centre O , ∢AOB = 54° . The point C is on the major arc. What size is ∢ACB ?
Answer and solution
The answer: 27 °
Both angles stand on the arc AB , so the inscribed angle is half the central one: ∢ACB = 54° ÷ 2 = 27° .
The angle in a semicircle
For example: AB is a diameter of a circle, and C is a point on the circle. ∢CAB = 26° . What size is ∢CBA ?
Answer and solution
The answer: 64 °
∢ACB stands on the diameter, so ∢ACB = 90° . The angles of a triangle add up to 180°, so ∢CBA = 180° − 90° − 26° = 64° .
Chords and their distance from the centre
For example: A circle has a radius of 5 cm, and the chord AB is 4 cm from the centre O . How long is the chord?
Answer and solution
The answer: 6 cm
In triangle OMA , by Pythagoras’ theorem AM = √(5² − 4²) = √9 = 3 . The chord is twice that: AB = 2 × 3 = 6 .
Cyclic quadrilaterals
For example: The quadrilateral ABCD is inscribed in a circle, and ∢A = 60° . What size is ∢C ?
Answer and solution
The answer: 120 °
∢C is opposite ∢A : ∢C = 180° − 60° = 120° .
Tangents to a circle
For example: A circle has a radius of 4 cm, and the point P is 5 cm from its centre O . How long is the tangent PA ?
Answer and solution
The answer: 3 cm
∢OAP is a right angle. By Pythagoras’ theorem PA = √(5² − 4²) = √9 = 3 .
A quadrilateral round a circle
For example: A circle is inscribed in the quadrilateral ABCD . In centimetres, AB = 12, BC = 12, CD = 8 . How long is DA ?
Answer and solution
The answer: 8 cm
The sums of opposite sides are equal, AB + CD = BC + DA , so DA = 12 + 8 − 12 = 8 .
The angle between a tangent and a chord
For example: The line AE touches the circle with centre O at A , and AB is a chord. The point C is on the major arc, and ∢ACB = 27° . What size is ∢BAE ?
Answer and solution
The answer: 27 °
∢BAE and the inscribed angle ∢ACB stand on the same arc AB , so they are equal: ∢BAE = 27° .
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).