iGCSE Extended Mathematics CAIE
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Start revising this course →Content Overview
107 topics in 7 modules
☑️ Circle Theorems 12 topics
- Angles in a Semicircle (90 Degrees)
- Angles at the Centre and at the Circumference
- Angles in the Same Segment
- Cyclic Quadrilaterals
- Tangent & Radius Theorem
- Alternate Segment Theorem
- Angles in a Complete Turn (360 Degrees)
- Radii and Chords
- Perpendicular Chord Theorem
- Theorem on Lengths of Tangents from an External Point
- Identifying and Applying Theorems in Problem Solving
- Proofs of Circle Theorems
☑️ Algebra 17 topics
- Algebra Basics
- Algebraic Functions
- Algebraic Proportion
- Completing the Square
- Expanding Brackets
- Factorising
- Factorising Quadratics
- Graphical Inequalities
- Inequalities
- Making Formulas from Words
- Powers
- Rearranging Formulas
- Sequences
- Simultaneous Equations
- Solving Equations
- Solving Equations Using Graphs
- The Quadratic Formula
☑️ Geometry and Measures 21 topics
- 3D Shapes
- Area and Volume Conversions
- Bearings
- Circle Geometry
- Circles
- Congruence
- Geometry
- Geometry Problems
- Parallel Lines
- Perimeter and Area
- Polygons
- Scale Drawings
- Similarity
- Speed, Density and Pressure
- Surface Area and Nets
- Symmetry
- The Four Transformations
- Time
- Triangle Constructions
- Unit Conversions
- Volume
☑️ Graphs, Functions and Calculus 14 topics
- Coordinates
- Differentiation
- Distance-Time Graphs
- Finding the Gradient
- Functions
- Gradients of Real-Life Graphs
- Harder Graphs
- Parallel and Perpendicular Lines
- Plotting Straight-Line Graphs
- Quadratic Graphs
- Real-Life Graphs
- Straight-Line Graphs
- Trig Graphs
- y = mx + c
☑️ Number 18 topics
- Bounds
- Calculator Buttons
- Compound Growth and Decay
- Fractions and Recurring Decimals
- Fractions, Decimals and Percentages
- LCM and HCF
- Multiples, Factors and Prime Factors
- Order of Operations
- Percentages
- Place Value and Ordering Numbers
- Powers and Roots
- Prime Numbers
- Proportion
- Ratios
- Rounding Numbers
- Sets
- Standard Form
- Venn Diagrams
☑️ Probability and Statistics 18 topics
- Box and Whisker Plots
- Collecting Data
- Comparing Data Sets
- Cumulative Frequency
- Frequency Tables
- Grouped Frequency Tables
- Histograms
- Interpreting Data
- Listing Outcomes and Expected Frequency
- Mean, Median, Mode and Range
- Pie Charts
- Probability Basics
- Probability from Venn Diagrams
- Relative Frequency
- Scatter Diagrams
- Simple Charts
- The AND/OR Rules
- Tree Diagrams
☑️ Pythagoras, Trigonometry and Vectors 7 topics
- 3D Pythagoras
- 3D Trigonometry
- Pythagoras’ THeorem
- Sin, Cos and Tan for Larger Angles
- The Sine and Cosine Rules
- Trigonometry - Sin, Cos and Tan
- Vectors
iGCSE Extended Mathematics CAIE Revision Content
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iGCSE Extended Mathematics CAIE - Circle Theorems - Angles in a Semicircle (90 Degrees) Content Preview
Circle Theorems
Angles in a Semicircle (90 Degrees)
Angles in a Semicircle
Basic Principles
- The semicircle is a special case among the circle theorems. It states that the angle subtended at the circumference by a semicircle is always a right angle, or 90 degrees.
- This theorem is also known as the angle in a semi circle is 90 degrees theorem.
- The angle in the semicircle theorem is applicable irrespective of where on the circumference the angle is placed.
Understanding the Theorem
- If you have a circle with centre ‘O’ and a diameter from any point on its circumference, a semicircle is formed. If another point is chosen on the circumference and lines are drawn from it to the diameter's ends, a right-angled triangle is formed within the semicircle.
- The angle formed at the chosen point is always a right angle.
- The theorem applies to all diameters and respective subtended angles on the circle, so it is not specific to any single diameter.
Using The Theorem in Problem Solving
- If a problem requires the calculation of an angle and you're given a semicircle (or you can draw a diameter to create one), it might be useful to consider the angle in a semicircle theorem.
- Remember, in a semicircle, the angle subtended at the circumference is always 90 degrees. So if your triangle or shape lies inside a semicircle and one of its sides is a diameter, then you have a right-angled triangle.
- This could help not only to find the angles involved more conveniently but it could also bring the application of Pythagoras’ theorem or the trigonometric functions into play since the triangle is right-angled.
Tangible Examples
- A real-life example of angles in a semicircle theorem could be seen in a pizza slice. If you slice a pizza across its centre to form a half circle, cut a piece from the crust to the other edge, you create a right-angled pizza slice!
- Architecturally, dome structures may implement semicircular arcs. If a brace or beam is placed from one end to another, any additional crossbeam forming an angle with the diameter would be at 90 degrees.
Understanding these principles and putting them to good use will help uncomplicate many geometrical problems involving circles and semicircles. Practice consistently to gain a good grasp.
Question: In a circle with center O and a diameter AB, a point P is marked on the circumference such that AP and BP form two radii. If a point C is marked anywhere on the circumference of the semicircle formed by the diameter AB, what is the measure of the angle ACB?
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