Circle Theorems

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课程笔记

What Are Circle Theorems?

Circle theorems deal with angles formed by lines drawn within and connected to a circle.
Key parts of a circle: the radius, diameter, chord, tangent, arc, sector and segment.
You may also need basic facts about angles, triangles, quadrilaterals and parallel lines.

Parts of a circle

Angle in a Semicircle

The angle in a semicircle is always 90°.
The triangle must have its vertices on the circumference and one side as the diameter.
The 90° angle is opposite the diameter.
This theorem is also known as:×The angle subtended by a diameter is a right angle×.
Use Pythagoras' Theorem if a side length is needed.

Angle in a semicircle

Tangent & Radius

A tangent is a straight line that touches the circle at exactly one point.
The radius drawn to the point of tangency is perpendicular to the tangent.
The angle between a radius and a tangent is 90°.
When explaining, state:×A radius and a tangent meet at right angles×.
Ensure the line from the centre is a radius, not a chord.

Tangent and radius

Key Exam Tips

Mark the 90° angle on the diagram as soon as you spot the arrangement.
Use the fact that angles in a triangle sum to 180° to find missing angles.
In an isosceles triangle formed by two radii, base angles are equal.
Always use the correct keywords in your reasoning.

幻灯片

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练习题

免费预览——35 题中的 8 题。注册即可查看全部。

1.X, Y and Z are points on the __________ of the circle, centre O.
Easy
- Acircumference
- Bdiameter
- Cradius
- Dtangent
2.A, B and C are points on the circle, centre O. AC is a diameter. Give a reason why angle ABC is 90°.
Easy
- AThe angle in a semicircle is 90°
- BA radius and a tangent are perpendicular
- CAngles at the centre are twice angles at the circumference
- DOpposite angles in a cyclic quadrilateral sum to 180°
3.B and D are points on the circumference of a circle, centre O. AC is a straight line touching the circle at B only and BD is a straight line through O. Complete the statement: Angle ABD = ______ because ______.
Easy
- A90°; the angle between a tangent and a radius is 90°
- B90°; the angle in a semicircle is 90°
- C180°; angles on a straight line sum to 180°
- D0°; the tangent is parallel to the radius
4.A, B and C are points on the circumference of a circle, centre O. The straight line DE touches the circle at B. Write down the mathematical name for the line DE.
Easy
- ATangent
- BChord
- CRadius
- DDiameter
5.A, B and C are points on the circumference of a circle, centre O. Write down the mathematical name for the straight line AC.
Easy
- AChord
- BDiameter
- CRadius
- DTangent
6.A, B and C are points on a circle. AC is a diameter of the circle. Angle ACB = 53°. Find the value of a, the size of angle BAC.
Easy
- A37°
- B53°
- C90°
- D127°
7.P and Q are points on the circle, centre O. APB is a tangent to the circle at P. Angle OPB = 90°. Angle QPB = 53°. Find angle POQ (x).
Medium
- A106°
- B74°
- C53°
- D127°
8.A circle, centre O, has diameter EF. D is a point on the circumference. Angle DFE = 63°. Find angle DEF.
Medium
- A27°
- B63°
- C90°
- D117°

复习笔记

## Circle Theorems – Quick Revision ### 1. Angle in a Semicircle - **Theorem:** The angle in a semicircle is **90°**. - **When to use:** A triangle has its vertices on the circle and one side is a **diameter**. - The right angle is always opposite the diameter. - **Key phrase:** "The angle in a semicircle is 90°." ### 2. Tangent & Radius - **Theorem:** A radius and a tangent meet at **right angles (90°)**. - **When to use:** A tangent touches the circle at a point; the radius drawn to that point is perpendicular. - **Key phrase:** "A radius and a tangent meet at right angles (90°)." ### Tips for Exams - As soon as you spot a diameter with a triangle on the circumference, mark the angle opposite the diameter as 90°. - If you see a tangent, check if it meets a radius and mark the 90° angle. - Use basic angle facts (triangle sum = 180°, isosceles triangle properties) to find missing angles. ### Common Question Types - Find missing angles using the two theorems above. - Give reasons using the exact keywords. - Identify chords, diameters, tangents, radii. ### Key Facts - A diameter goes through the centre. - A radius is from centre to circumference. - A tangent touches the circle at exactly one point. - A chord is a straight line joining two points on the circumference.

Angle in a semicircle: The angle subtended by a diameter at any point on the circumference is 90°.
Tangent: A straight line that touches the circle at exactly one point.
Radius: A straight line from the centre of the circle to any point on the circumference.
Diameter: A straight line passing through the centre of the circle, connecting two points on the circumference.
Chord: A straight line joining two points on the circumference of a circle.

闪卡

What is the angle in a semicircle?

The angle in a semicircle is 90°.

State the circle theorem about a radius and a tangent.

A radius and a tangent meet at right angles (90°).

In the diagram, A, B, C are on a circle with AC as diameter. Why is angle ABC = 90°?

Angle in a semicircle is 90°.

What is the mathematical name for line DE that touches the circle at B?

Tangent.

Complete: Angle ABD = 90° because ...

the angle between a tangent and a radius is 90°.

What is the mathematical name for line AC in a circle where A and C are on the circumference and O is the centre?

Diameter (if it passes through O) or chord (if not).

In the diagram, points E, F, G lie on a circle with centre O. JGH is a tangent. Find x and y.

x = 90° (tangent-radius), y = 90° (angle in semicircle if EG is diameter).

The diagram shows vertices of a triangle on a circle with centre O. Find b and give reason.

b = 90° because angle in a semicircle is 90° (if side is diameter).

Points A, B, C on circle centre O. PBR is tangent, angle BAC = 37°. Find j and k.

j = 90° (tangent-radius), k = 53° (angles in triangle).

A, B, C on circle with diameter AB. Tangent at A. Find angle BAC.

Angle BAC = 90° (tangent-radius).

In the diagram, circle centre O with diameter PQ. R on circumference. PR = 5.2 cm, PR = 1/3 PQ. Why is angle PRQ = 90°?

Angle in a semicircle is 90°.

P and Q on circle centre O. APB tangent at P. Name line PQ.

Chord.

Explain why angle OPB = 90°.

The angle between a tangent and a radius is 90°.

Circle centre O with diameter CE. A, B, C, D, E on circumference. Find x and give reason.

x = 90° because angle in a semicircle is 90°.

Draw a tangent to a circle at point A.

A straight line that touches the circle at A only.

Circle centre O, diameter EF. Angle DFE = 63°. Find angle DEF.

Angle DEF = 27° (angle in semicircle is 90°, then triangle sum).

EF = 12 cm, angle DFE = 63°. Calculate DF.

DF = 12 sin 63° ≈ 10.7 cm.

K, L, M on circle. KS tangent at K. KM diameter. Triangle KLM is isosceles. Find z.

z = 45° (angle in semicircle 90°, isosceles gives base angles 45°).

Line AC touches circle at B. OB parallel to DC. Angle OAB = 49°. Name line OB.

Radius.

Why is angle ABO = 90°?

The angle between a tangent and a radius is 90°.

Find angle AOB given angle OAB = 49° and angle ABO = 90°.

Angle AOB = 41° (triangle sum).

Why is angle ADC = angle AOB?

Corresponding angles (OB parallel to DC).

Triangle AOB is _______ to triangle ADC.

Similar.

AB = 5.4 cm, angle OAB = 49°, angle ABO = 90°. Calculate OB.

OB = 5.4 tan 49° ≈ 6.21 cm.

AB = 5.4 cm, angle OAB = 49°, angle ABO = 90°. Calculate OA.

OA = 5.4 / cos 49° ≈ 8.23 cm.

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