Circle and related theorems

Unit 9 Circle and related theorems

Key unit competence: Use properties of circle and circle theorems in constructing mathematical arguments and solving related problems.

Key ideas

• Parts: centre, radius, diameter $=2r$, chord, arc, sector, segment, tangent. Circumference $C=2\pi r$, area $A=\pi r^2$.
• Arc length for central angle $\theta$ (degrees): $l=\frac{\theta }{360}\times 2\pi r$. Sector area: $\frac{\theta }{360}\times \pi r^2$.
• Chords: the perpendicular from the centre to a chord bisects the chord; equal chords are equidistant from the centre.
• Theorem 1: the angle at the centre is twice the angle at the circumference subtended by the same arc.
• Theorem 2: the angle in a semicircle is a right angle.
• Theorem 3: angles in the same segment (subtended by the same arc) are equal.
• Theorem 4: opposite angles of a cyclic quadrilateral add up to $180^{\circ }$; an exterior angle equals the interior opposite angle.
• Theorem 5: a tangent is perpendicular to the radius at the point of contact; the two tangents from an external point are equal in length.

Formulas to know

Common misconceptions

• "The angle at the circumference is twice the angle at the centre." No: the angle at the centre is twice the angle at the circumference.
• "Opposite angles of any quadrilateral add to $180^{\circ }$." Only for a cyclic quadrilateral (vertices on a circle).

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. State the five circle theorems of this unit.

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   (1) Angle at centre = twice angle at circumference; (2) angle in a semicircle = $90^{\circ }$; (3) angles in the same segment are equal; (4) opposite angles of a cyclic quadrilateral add to $180^{\circ }$; (5) tangent ⟂ radius, and tangents from an external point are equal.

   Common mistake: Reversing theorem 1.

2. A wheel of diameter 70 cm turns 100 times. How far does it travel ($\pi =\frac{22}{7}$)?

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   Circumference 220 cm; $100\times 220=22000$ cm $=220$ m.

   Common mistake: Using the radius as the diameter.

3. Find the arc length and the area of a sector of radius 14 cm and angle $45^{\circ }$ ($\pi =\frac{22}{7}$).

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   $l=\frac{1}{8}\times 88=11$ cm; area $=\frac{1}{8}\times 616=77$ cm².

   Common mistake: Using the diameter in the formula.

4. Prove that the angle in a semicircle is a right angle using theorem 1.

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   The diameter subtends a straight angle of $180^{\circ }$ at the centre, so the angle at the circumference is $\frac{180^{\circ }}{2}=90^{\circ }$.

   Common mistake: Assuming the triangle is isosceles.

5. A chord is 5 cm from the centre of a circle of radius 13 cm. Find its length.

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   Half chord $=\sqrt{169-25}=12$, chord $=24$ cm.

   Common mistake: Forgetting to double the half chord.

6. From a point $P$ 10 cm from the centre $O$, a tangent is drawn to a circle of radius 6 cm. Find the tangent length.

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   $\sqrt{100-36}=8$ cm.

   Common mistake: Adding the squares.

7. $ABCD$ is cyclic with $\angle A=3x$ and $\angle C=2x+20^{\circ }$. Find $x$ and the two angles.

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   $5x+20=180$, $x=32^{\circ }$; $\angle A=96^{\circ }$, $\angle C=84^{\circ }$.

   Common mistake: Setting the angles equal.

8. $AB$ is a diameter of a circle, $C$ on the circle, $AC=6$ cm, $BC=8$ cm. Find the radius.

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   $\angle C=90^{\circ }$, $AB=10$ cm, radius 5 cm.

   Common mistake: Not recognising the right angle in a semicircle.

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Find the area of a circle of radius 7 cm ($\pi =\frac{22}{7}$).	a. 154 cm²
   2. On the same arc, the angle at the centre is $(2x+10{)}^{\circ }$ and the angle at the circumference is $50^{\circ }$. Find $x$.	b. $70^{\circ }$
   3. The angle at the centre is $140^{\circ }$. A point on the major arc sees the chord at:	c. $65^{\circ }$
   4. A tangent makes an angle of $65^{\circ }$ with a chord at the point of contact. Find the angle subtended by the chord in the alternate segment.	d. 45

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   1 → a (154 cm²) · 2 → d (45) · 3 → b ($70^{\circ }$) · 4 → c ($65^{\circ }$)

   • $\frac{22}{7}\times 49$.
   • Step 1. The angle at the centre is twice the angle at the circumference: $2x+10=100$.
     Step 2. $2x=90$, so $x=45$.
   • Half of $140^{\circ }$.
   • Step 1. Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
     Step 2. So the angle is $65^{\circ }$.

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   10 cm  ·  $50^{\circ }$  ·  $35^{\circ }$  ·  $200^{\circ }$
   1. The angle at the centre subtended by an arc is $100^{\circ }$. The angle at the circumference subtended by the same arc is .
   2. Two angles at the circumference stand on the same arc. One is $35^{\circ }$. The other is .
   3. A chord of 16 cm is 6 cm from the centre. The radius is .
   Show answer

   1: $50^{\circ }$ · 2: $35^{\circ }$ · 3: 10 cm

   • Half of the angle at the centre.
   • Angles in the same segment are equal.
   • $\sqrt{64+36}$.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. A chord of length 24 cm lies in a circle of radius 13 cm. How far is it from the centre?	a. $85^{\circ }$
   2. An arc of length 11 cm lies on a circle of radius 7 cm. Find the angle at the centre ($\pi =\frac{22}{7}$).	b. $90^{\circ }$
   3. Two tangents are drawn from an external point $P$ to a circle. They are:	c. equal in length
   4. In a cyclic quadrilateral $ABCD$, $\angle B=95^{\circ }$. Then $\angle D=$	d. 5 cm

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   1 → d (5 cm) · 2 → b ($90^{\circ }$) · 3 → c (equal in length) · 4 → a ($85^{\circ }$)

   • Step 1. The perpendicular from the centre bisects the chord: half-chord 12 cm.
     Step 2. Distance $=\sqrt{13^2-12^2}=5$ cm.
   • Step 1. Arc $=\frac{\theta }{360^{\circ }}\times 2\pi r=\frac{\theta }{360^{\circ }}\times 44$.
     Step 2. $\frac{\theta }{360^{\circ }}\times 44=11\Rightarrow \theta =90^{\circ }$.
   • Tangents from an external point are equal.
   • $180^{\circ }-95^{\circ }$.

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   is a tangent  ·  $9\pi$ cm²  ·  $70^{\circ }$  ·  bisects the chord
   1. In a cyclic quadrilateral one angle is $110^{\circ }$. The opposite angle is .
   2. Sector area for $r=6$ cm and $\theta =90^{\circ }$ is .
   3. The perpendicular from the centre to a chord .
   Show answer

   1: $70^{\circ }$ · 2: $9\pi$ cm² · 3: bisects the chord

   • Opposite angles are supplementary.
   • $\frac{1}{4}\times 36\pi$.
   • It meets the chord at its midpoint.

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