# Circles: angles, chords and tangents · Grade 10 · 5 units

> Explanation, worked examples and free practice following the Israeli curriculum.

In Circles: angles, chords and tangents (Grade 10 · 5 units) 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

- Example: (A circle, a chord, and the central and inscribed angles standing on it.) In a circle with centre O , ∢AOB = 54° . The point C is on the major arc. What size is ∢ACB ?
- Answer: 27 °
- Solution: 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

- Example: (A circle, a diameter, and a point on the circle joined to both ends of it.) AB is a diameter of a circle, and C is a point on the circle. ∢CAB = 26° . What size is ∢CBA ?
- Answer: 64 °
- Solution: ∢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

- Example: (A circle, a chord, the perpendicular from the centre to it, and the radius to its end.) 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: 6 cm
- Solution: In triangle OMA , by Pythagoras’ theorem AM = √(5² − 4²) = √9 = 3 . The chord is twice that: AB = 2 × 3 = 6 .

### Cyclic quadrilaterals

- Example: (A quadrilateral inscribed in a circle.) The quadrilateral ABCD is inscribed in a circle, and ∢A = 60° . What size is ∢C ?
- Answer: 120 °
- Solution: ∢C is opposite ∢A : ∢C = 180° − 60° = 120° .

### Tangents to a circle

- Example: (A circle, a tangent to it from a point outside, and the radius to the point of contact.) 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: 3 cm
- Solution: ∢OAP is a right angle. By Pythagoras’ theorem PA = √(5² − 4²) = √9 = 3 .

### A quadrilateral round a circle

- Example: (A quadrilateral drawn round a circle: each side touches the circle.) A circle is inscribed in the quadrilateral ABCD . In centimetres, AB = 12, BC = 12, CD = 8 . How long is DA ?
- Answer: 8 cm
- Solution: 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

- Example: (A circle, a tangent at a point of it, a chord from that point, and an inscribed angle standing on the chord.) 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: 27 °
- Solution: ∢BAE and the inscribed angle ∢ACB stand on the same arc AB , so they are equal: ∢BAE = 27° .

## Not on the site yet

- Proofs of the circle theorems
- Constructions with a ruler and compasses

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).

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