Circle theorems (OCR GCSE Maths): Revision Notes

Example: If the angle at the circumference is $x$, then the angle at the centre is $2x$ Diagram Explanation: Imagine an arc of a circle with two endpoints. Draw two lines from these endpoints to the centre of the circle, creating an angle at the centre, $2x$. Now draw lines from the same two endpoints to any point on the circumference, creating an angle $x$. The angle at the centre ($2x$) is twice the angle at the circumference ($x$).

Example:

• If you have a triangle inscribed in a circle where one side is the diameter of the circle, the angle opposite the diameter will always be $90\degree$.

Diagram Explanation: Consider a semi-circle where the diameter is the base of a triangle. The two ends of the diameter are connected to any point on the circumference, forming a triangle. The angle opposite the diameter, at the circumference, will always be $90\degree$.

Note: This theorem is actually a special case of Theorem $1$. When the angle at the centre is $180\degree$ (as the diameter passes through the centre), the angle at the circumference must be half of this, hence $90\degree$

Example:

• In a circle, if two different angles are subtended by the same chord and lie in the same segment, then these angles are equal, i.e., $∠a=∠b.$

Diagram Explanation: Consider two points on the circumference of a circle, $A$ and $B$, connected by a chord. Now draw two lines from points $A$ and $B$ to any point on the same segment of the circle, forming two angles. According to this theorem, these two angles are equal.

Example:

• If you have a cyclic quadrilateral $ABCD ,then \ :highlight[∠A+∠C=180\degree] and\ :highlight[∠B+∠D=180\degree]$.

Diagram Explanation: Consider a quadrilateral inscribed in a circle, where all the vertices ($A, B, C, D$) lie on the circumference. According to this theorem, the sum of opposite angles $∠A+∠C and ∠B+∠D$ will each add up to $180\degree$.

Note:

• Like any other quadrilateral, the sum of all interior angles in a cyclic quadrilateral is still $360\degree$.

Example:

• Given a circle with centre $O$ and a tangent $TP$ touching the circle at point $P$, the angle between OP (radius) and $TP$(tangent) is $90\degree$.

Diagram Explanation:

• Consider a circle with a tangent touching it at point $P$. Draw the radius $OP$ from the centre $O$ to the point of contact. The angle $∠OPT$ formed between the radius and the tangent is $90\degree$.

Example:

• If $TQ$ is a tangent to the circle at point $Q$and $QR$ is a chord, then the angle $∠TQR$ between the tangent and the chord is equal to the angle subtended by the chord $QR$ at the circumference in the opposite segment.

Diagram Explanation:

• Consider a circle with a tangent $TQ$ touching the circle at $Q$ and a chord $QR$ drawn inside the circle. The angle $∠TQR$ formed between the tangent and the chord is equal to the angle subtended by $QR$ on the opposite side of the circle.

Example:

• Consider a point $P$ outside the circle. Two tangents $PA$ and $PB$ touch the circle at points $A$ and $B$ respectively. According to Theorem $7$, the lengths $PA$ and $PB$ are equal, i.e., $PA=PB.$

Diagram Explanation:

• Imagine a point $P$ outside the circle. Draw two tangents $PA$ and $PB$ from $P$ to touch the circle at points $A$ and $B$. The lengths of these tangents will be equal, so $PA=PB$.

Example 1:

• Given: $∠BAC=75\degree$ and $∠ABC=35\degree$.
• Find $∠ACB=x$ and $∠BOC=y$.

Solution:

• Use the angle sum of a triangle:

• For $y$, since $∠BDA$ subtends the same arc as $∠BCA$:

Example 2:

• Given: $∠A=40\degree$. Find $∠B$ and $∠C$.

Solution:

• Since $∠B$ is opposite the diameter, it is a right angle:

• Using the angle sum of a triangle:

📑Example 3:

• Given: $∠A=88\degree$. Find $∠B$.

  

• Solution:

• Since $∠A$ and $∠B$ are in the same segment:

📑Example 4:

• Given: A quadrilateral inside a circle with one angle $∠A=90\degree$. Find the opposite angle $∠C$.

• Solution:
• The sum of the opposite angles in a cyclic quadrilateral is $180°$:

📑Example 5:

• Given: $∠y=36\degree$, $∠x=24\degree$, and a cyclic quadrilateral is formed with one exterior angle.
• Find: The values of $x$, $y$, and $z$.

Solution:

1. Determine $y$:

• $y$ is found using the Alternate Segment Theorem:

2. Determine $x$:

• $x$ is found using the angle sum in a triangle:

3. Determine $z$=

• $z$ is found using the Cyclic Quadrilateral Theorem:

Thus, the solutions are:

• $y=36\degree$
• $x=120\degree$
• $z=60\degree$