Circle theorems II

1. Overview

Circle Theorems II focuses on three key symmetry properties of circles related to chords and tangents. These properties allow you to calculate unknown lengths and, less commonly, angles within circles. The core concepts involve understanding the relationships between the circle's centre, chords, perpendicular bisectors, and tangents drawn from external points. Mastery of these theorems often requires applying Pythagoras' Theorem and trigonometric ratios.

Key Definitions

• Chord: A straight line segment whose endpoints both lie on the circumference of a circle.
• Diameter: A specific chord that passes through the centre of the circle (the longest possible chord).
• Radius: The distance from the centre of the circle to any point on its circumference.
• Tangent: A straight line that touches the circumference of a circle at exactly one point.
• Perpendicular Bisector: A line that cuts another line exactly in half at a 90° angle.
• Equidistant: Being at an equal distance from a specific point or line.

Core Content

There are no specific Core-only objectives for this sub-topic. All learning objectives for Circle Theorems II are part of the Extended (Supplement) curriculum.

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Extended Content (Extended Only)

The following properties rely on the symmetry of the circle. Many problems involving these theorems will require you to use Pythagoras’ Theorem ($a^2 + b^2 = c^2$) or Trigonometry (SOH CAH TOA).

Property 1: Equal chords are equidistant from the centre

If two chords in the same circle have the same length, they must be the same distance from the centre of the circle. The distance is measured along the perpendicular from the centre to the chord.

Worked example 1 — Distance to equal chords

In a circle with centre $O$, chord $PQ = 8\text{ cm}$ and chord $RS = 8\text{ cm}$. The perpendicular distance from $O$ to $PQ$ is $5\text{ cm}$. Find the perpendicular distance from $O$ to $RS$.

• Step 1: Identify that $PQ = RS = 8\text{ cm}$.
• Step 2: Apply the theorem: equal chords are equidistant from the centre.
• Step 3: Therefore, distance $O$ to $RS = 5\text{ cm}$.

Worked example 2 — Radius and distance to equal chords

In a circle with centre $O$ and radius $10\text{ cm}$, two chords $AB$ and $CD$ are equal in length. The distance from the centre $O$ to chord $AB$ is $6\text{ cm}$. Calculate the length of chord $CD$.

• Step 1: Draw a perpendicular line from $O$ to $AB$, meeting $AB$ at point $M$. $OM = 6\text{ cm}$.
• Step 2: Draw radius $OA$. Triangle $OMA$ is a right-angled triangle.
• Step 3: Use Pythagoras' Theorem to find $AM$: $OA^2 = OM^2 + AM^2$ $10^2 = 6^2 + AM^2$ $100 = 36 + AM^2$ $AM^2 = 64$ $AM = \sqrt{64} = 8\text{ cm}$
• Step 4: Since $OM$ is perpendicular to $AB$, it bisects $AB$. Therefore, $AB = 2 \times AM = 2 \times 8 = 16\text{ cm}$.
• Step 5: Since $AB = CD$, then $CD = 16\text{ cm}$.

Property 2: The perpendicular bisector of a chord passes through the centre

Any line that cuts a chord in half at a 90° angle will always pass through the centre of the circle. Conversely, a radius that is perpendicular to a chord bisects (splits in half) that chord. This property is often used to find the centre of a circle if it is not given.

Worked example 3 — Finding the radius

A chord of length $8\text{ cm}$ is $3\text{ cm}$ from the centre $O$ of a circle. Calculate the radius of the circle.

• Step 1: The line from the centre to the chord is a perpendicular bisector.
• Step 2: Half the chord length = $8 \div 2 = 4\text{ cm}$.
• Step 3: Form a right-angled triangle where the legs are $4\text{ cm}$ and $3\text{ cm}$, and the hypotenuse is the radius ($r$).
• Step 4: Use Pythagoras’ Theorem: $r^2 = 4^2 + 3^2$ $r^2 = 16 + 9$ $r^2 = 25$ $r = \sqrt{25} = 5\text{ cm}$

Worked example 4 — Finding the distance to the chord

A circle has a radius of $17\text{ cm}$. A chord within the circle is $30\text{ cm}$ long. Find the perpendicular distance from the centre of the circle to the chord.

• Step 1: Draw a radius from the centre $O$ to one end of the chord, $A$. This forms a right-angled triangle.
• Step 2: The perpendicular distance from the centre to the chord bisects the chord. Half the chord length is $30 \div 2 = 15\text{ cm}$.
• Step 3: Let the perpendicular distance be $d$. Use Pythagoras' Theorem: $17^2 = 15^2 + d^2$ $289 = 225 + d^2$ $d^2 = 289 - 225$ $d^2 = 64$ $d = \sqrt{64} = 8\text{ cm}$

Property 3: Tangents from an external point are equal in length

If two tangents are drawn to a circle from the same external point, the lengths from that point to the points of contact on the circle are equal. This creates a shape with two congruent right-angled triangles. The line joining the external point to the centre of the circle bisects the angle between the two tangents.

Worked example 5 — Tangent length

Point $T$ is $13\text{ cm}$ from the centre $O$ of a circle with radius $5\text{ cm}$. A tangent is drawn from $T$ to the circle at point $A$. Calculate the length of the tangent $TA$.

• Step 1: Recognize that the radius $OA$ is perpendicular to the tangent $TA$ (Angle $OAT = 90^\circ$).
• Step 2: Use Pythagoras’ Theorem ($a^2 + b^2 = c^2$), where $OT$ is the hypotenuse.
• Step 3: Calculate length: $TA^2 + 5^2 = 13^2$ $TA^2 + 25 = 169$ $TA^2 = 169 - 25$ $TA^2 = 144$ $TA = \sqrt{144} = 12\text{ cm}$

Worked example 6 — Finding the angle

From point $P$ outside a circle with centre $O$, two tangents $PA$ and $PB$ are drawn to the circle, where $A$ and $B$ are the points of tangency. If $\angle APB = 40^\circ$, find the angle $\angle OAP$ and $\angle AOB$.

• Step 1: Recognize that $PA = PB$ (tangents from an external point are equal).
• Step 2: $\angle OAP = \angle OBP = 90^\circ$ (tangent meets radius at $90^\circ$).
• Step 3: Consider quadrilateral $OAPB$. The sum of angles in a quadrilateral is $360^\circ$.
• Step 4: Calculate $\angle AOB$: $\angle AOB = 360^\circ - \angle OAP - \angle OBP - \angle APB$ $\angle AOB = 360^\circ - 90^\circ - 90^\circ - 40^\circ$ $\angle AOB = 140^\circ$

Key Equations

These formulas are not provided on the IGCSE formula sheet and must be memorised/applied:

Pythagoras’ Theorem: $a^2 + b^2 = c^2$ (Used for chord and tangent lengths).

Trigonometric Ratios:

• $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

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Common Mistakes to Avoid

• ❌ Wrong: Assuming that any line from the centre of the circle to a chord bisects the chord.
• ✅ Right: The line from the centre to the chord must be perpendicular to the chord to bisect it.
• ❌ Wrong: Forgetting to halve the chord length when using Pythagoras' Theorem to find the distance from the centre to the chord or the radius.
• ✅ Right: Always draw a perpendicular line from the centre to the chord and remember to use half the chord length in your calculations.
• ❌ Wrong: Assuming that the angle between a tangent and a line from the external point to the centre of the circle is 90°.
• ✅ Right: The tangent meets the radius at 90°. Draw the radius to the point of tangency to form a right angle.
• ❌ Wrong: Not recognizing that tangents from an external point create congruent triangles.
• ✅ Right: Look for congruent triangles formed by the tangents, radii, and the line joining the external point to the centre. This can help you find equal angles and side lengths.

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Exam Tips

• Command Words: "Show that" or "Calculate". If the question says "Give reasons for your answer," you must state the relevant circle theorem (e.g., "tangents from an external point are equal in length").
• Hidden Right Angles: Always look for opportunities to draw a radius to the point of tangency to create a right-angled triangle.
• Isosceles Triangles: Remember that two radii of a circle are always equal. Look for isosceles triangles formed by radii and chords. This means the base angles are equal.
• Calculator vs Non-Calculator: In calculator papers, you may get non-integer results. Round your final answers to 3 significant figures unless specified otherwise, but keep full values on your calculator during working.
• Typical Values: Be familiar with Pythagorean triples like (3, 4, 5), (5, 12, 13), and (8, 15, 17) as these frequently appear in circle questions.
• Diagrams: Always draw on the diagram provided in the question. Add radii, perpendicular bisectors, and right angles to help visualize the problem.