Begin by joining the point O to the point A and marking all the radiuses as equal lengths on the diagram.

0XZ7dlRs_4-8-edexcel-gcse-vh-q2-diagram-1

This creates two isosceles triangles, triangle AOB and triangle AOC.
The base angles of isosceles triangles are equal, mark these equal angles on the diagram, labelling the angles in one triangle with one variable and the angles in the other triangle with a different variable.

Let angle $O A B = x$ and angle $O A C = y$ .
Then angle $B A C = x + y$ .

$O \hat{A} B = O \hat{B} A = x$

$O \hat{A} C = O \hat{C} A = y$

4-8-edexcel-gcse-vh-q2-diagram-2

Write angle BAC in terms of $x$ and $y$ .

$B \hat{A} C = x + y$

[1]

Angles in a triangle add up to 180°, use this to find the angles AOC and BOC.

$A \hat{O} B = 180 - x - x = 180 - 2 x$

$A \hat{O} C = 180 - y - y = 180 - 2 y$

Angles at a point add up to 360°, use this to find an expression for angle BOC in terms of $x$ and $y$ .

$B \hat{O} C + A \hat{O} B + A \hat{O} C = 360 ° B \hat{O} C + (180 - 2 x) + (180 - 2 y) = 360$

[1]

Simplify.

$\begin{array}{rcl} B \hat{O} C + 180 - 2 x + 180 - 2 y & = & 360 \\ B \hat{O} C + 360 - 2 x - 2 y & = & 360 \\ B \hat{O} C - 2 (x + y) & = & 0 \\ B \hat{O} C & = & 2 (x + y) \end{array}$

[1]

We have already shown that $B \hat{A} C = x + y$ .

$\begin{array}{rcl} B \hat{O} C & = & 2 (x + y) = 2 B \hat{A} C \end{array}$

The base angles of an isosceles triangle are equal
Angles at a point add up to 360°

Therefore Angle BOC = 2(Angle BAC )All reasons given clearly [1]

$□$	$A B$ is parallel to $D C$
$□$	$A C$ bisects angle $B C D$
$□$	$A C$ bisects angle $B A D$

Circle Theorems (OCR GCSE Maths)

Exam Questions

1. Number Operations & Integers

Number Toolkit

Prime Factors, HCF & LCM

2. Fractions, Decimals & Percentages

Fractions

Percentages

Working with FDP

3. Indices & Surds

Powers, Roots & Standard Form

Surds

4. Approximation & Estimation

Rounding, Estimation & Bounds

Using a Calculator

5. Ratio, Proportion & Rates of Change

Ratio Toolkit

Ratio Problem Solving

Direct & Inverse Proportion

Exchange Rates & Best Buys

Simple & Compound Interest, Growth & Decay

6. Algebra

Algebra Toolkit

Algebraic Roots & Indices

Expanding Brackets

Factorising

Completing the Square

Algebraic Fractions

Rearranging Formulae

Algebraic Proof

Linear Equations

Solving Quadratic Equations

Solving Inequalities

Simultaneous Equations

Iteration

Forming & Solving Equations

Functions

Sequences

7. Graphs of Equations & Functions

Coordinate Geometry

Linear Graphs y = mx + c

Graphing Inequalities

Graphs of Functions

Equation of a Circle

Estimating Gradients & Areas under Graphs

Real-Life Graphs

Transformations of Graphs

8. Basic Geometry

Geometry Toolkit

Angles in Polygons & Parallel Lines

Circle Theorems

9. Congruence & Similarity

Congruence, Similarity & Geometrical Proof

Area & Volume of Similar Shapes

Vectors

Transformations

10. Mensuration

Standard & Compound Units

Bearings, Scale Drawing, Constructions & Loci

Area & Perimeter

Circles, Arcs & Sectors

Volume & Surface Area

Right-Angled Triangles - Pythagoras & Trigonometry

Sine, Cosine Rule & Area of Triangles

3D Pythagoras & Trigonometry

11. Probability

Probability Toolkit

Simple Probability Diagrams

Tree Diagrams

Combined & Conditional Probability

12. Statistics

Statistics Toolkit

Statistical Diagrams

Histograms

Cumulative Frequency & Box Plots

Scatter Graphs & Correlation