Circle Theorems

விளையாடிக் கற்றுக்கொள்ளுங்கள்

ஆற்றல் சம்பாதிக்க இந்த கேள்விகளுக்குப் பதிலளியுங்கள், பின்னர் மீன் பிடித்து ஆராயுங்கள். கணக்கு தேவையில்லை.

ஆசிரியர்களுக்கு: Circle Theorems (Maths [CIE], Core)-க்கான தயாரான பாட ஸ்லைடுகள், திருப்புதல் குறிப்புகள், வரைபடங்கள் — உங்கள் பாடத்தில் அவற்றைப் பயன்படுத்தவும், அல்லது மாணவர்கள் நேரலை விளையாட்டாக விளையாடும் ஊடாடும் வகுப்பு செயல்பாடாக தலைப்பை இயக்கவும்.

பாட குறிப்புகள்

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.

ஸ்லைடுகள்

Step through every slide for this topic — plus flashcards and revision notes — with a free account.

பதிவு செய்க — இலவசம்உள்நுழைக

பயிற்சி கேள்விகள்

இலவச முன்னோட்டம் — 35-இல் 8 கேள்விகள். அனைத்தையும் பார்க்க பதிவு செய்யவும்.

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°

மேலும் கேள்விகள்

9.K, L and M are points on a circle. KM is a diameter and triangle KLM is isosceles. Find angle KML (z).
Hard
- A45°
- B90°
- C30°
- D60°
10.What is the mathematical name for a line that touches a circle at exactly one point?
Easy
- ATangent
- BChord
- CRadius
- DDiameter
11.What is the angle between a radius and a tangent at the point of contact?
Easy
- A90°
- B180°
- C0°
- D45°
12.In the diagram, O is the centre of the circle. The radius is 5 cm and a tangent is drawn from point T to the circle, touching at P. If TP = 12 cm, find the distance OT.
Medium
- A13 cm
- B17 cm
- C7 cm
- D√119 cm
13.A, B, C are points on a circle with centre O. AC is a diameter. AB = 6 cm, BC = 8 cm. Find the radius of the circle.
Medium
- A5 cm
- B7 cm
- C10 cm
- D14 cm
14.Which circle theorem states that the angle in a semicircle is 90°?
Easy
- AAngle at centre theorem
- BAngle in a semicircle theorem
- CTangent-radius theorem
- DCyclic quadrilateral theorem
15.A circle has radius 5 cm. A tangent is drawn from an external point T, touching the circle at P. If OT = 13 cm, find TP.
Easy
- A8 cm
- B12 cm
- C√194 cm
- D18 cm
16.A, B, C are points on a circle. AC is a diameter. Angle ABC = 90°. If AB = 3 cm, BC = 4 cm, find the area of the circle.
Medium
- Aπ × (2.5)² cm²
- Bπ × 5² cm²
- Cπ × 6.25 cm²
- Dπ × 10 cm²
17.Points P, Q, R lie on a circle with centre O. PR is a diameter. Angle RPQ = 25°. Find angle PQR.
Medium
- A65°
- B90°
- C155°
- D25°
18.A, B and C are points on a circle. AC is a diameter. Angle BAC = 53°. Find angle ABC.
Easy
- A37°
- B53°
- C90°
- D127°
19.The angle in a semicircle is always 90°.
Easy
True or false?
20.Match each circle theorem to its correct statement.
Easy
- Angle in a semicircle
- Tangent and radius
- The angle is 90°
- They meet at right angles
21.A, B and C are points on a circle with centre O. AC is a diameter. Which angle is 90°?
Easy
- AAngle ABC
- BAngle BAC
- CAngle ACB
- DAngle AOC
22.Arrange the following steps to find the value of x, where AC is a diameter of a circle and angle BAC = 53°.
Medium
- State that angle ABC = 90° (angle in a semicircle)
- Write equation: x + 53 + 90 = 180
- Solve: x = 37°
23.Which of the following statements about a circle are true? (Select all that apply)
Medium
- AA radius and a tangent meet at 90°
- BThe angle in a semicircle is 180°
- CA diameter is a chord that passes through the centre
- DA tangent touches the circle at two points
24.Which of the following statements is correct?
Easy
- AThe angle in a semicircle is 90°.
- BThe angle in a semicircle is 180°.
- CThe angle in a semicircle is 45°.
- DThe angle in a semicircle is 60°.
25.A radius and a tangent at the point of contact are always perpendicular.
Easy
True or false?
26.Fill in the blank.
Easy
A line that touches a circle at exactly one point is called a ____.
27.A circle has a radius of 5 cm. A tangent is drawn from an external point T, touching the circle at P. If OT = 13 cm, find TP.
Easy
28.A, B, C lie on a circle with centre O. AC is a diameter. AB = 6 cm, BC = 8 cm. Find the radius of the circle.
Medium
29.Complete the sentence.
Easy
The angle between a radius and a tangent at the point of contact is ____ degrees.
30.A circle has a diameter of 28 cm. Find the radius of this circle.
Easy
31.P and Q are points on the circle, centre O. APB is a tangent to the circle at P. Explain why angle OPB is 90°.
Easy
32.Fill in the blanks to complete the circle theorem:
Easy
The angle in a ____ is ____ degrees.
33.Complete the circle theorem statement.
Easy
The angle between a ____ and a ____ at the point of contact is ____ degrees.
34.In the diagram, O is the centre of the circle. P and Q are points on the circumference. APB is a tangent to the circle at P. Angle QPB = 53°. Find angle POQ.
Medium
35.A circle has a diameter of 28 cm. Find the radius of this circle.
Easy
- A16 cm
- B13 cm
- C15 cm
- D14 cm

திருப்புதல் குறிப்புகள்

## 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.

ஃப்ளாஷ் கார்டுகள்

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.

Unlock all 35 questions, flashcards & more

இந்த தலைப்பிற்கான ஒவ்வொரு கேள்வியையும், ஸ்லைடுகளையும், ஃப்ளாஷ் கார்டுகளையும் மற்றும் திருப்புதல் குறிப்புகளையும் பார்க்க இலவச கணக்கை உருவாக்கவும்.

பதிவு செய்க — இலவசம்உள்நுழைக

முந்தைய தேர்வுகள்

இந்த தலைப்பிற்கான முந்தைய தேர்வு பயிற்சி விரைவில் வருகிறது.

விரைவில் வருகிறது