Geometry, circle theorems and similarity
Turn geometric information into a chain of justified angle and scale relationships.
What you will learn
- Combine circle and triangle properties.
- Use squared and cubed scale factors.
Name the reason for each angle
Angles in a triangle total 180°. Equal sides in an isosceles triangle have equal opposite angles. Circle results connect different parts of a diagram: for example, the angle between a tangent and chord equals the angle in the alternate segment.
Identify the particular angle before applying a theorem. A diagram can contain both an acute and a supplementary angle at the same intersection. Use the vertex order in three-letter angle names and the other geometric constraints to select the right one.
Different quantities scale differently
If two shapes are mathematically similar and their length scale factor is k, their area scale factor is k². For similar solids, the volume scale factor is k³. Work out whether the question gives lengths, areas or volumes before converting the ratio.
Worked example
Similar solids have lengths in the ratio 2:3. The smaller has surface area 40 cm². Find the larger surface area.
Show the worked solution
- The length scale factor from smaller to larger is 3/2.
- Square it for surface area: (3/2)²=9/4.
- Multiply 40 by 9/4.
Answer 90 cm²
Common mistakes
- An area ratio is not the same as a length ratio.
- A theorem name without identifying the relevant angles is not a complete explanation.
What can you explain now?
An isosceles triangle has two base angles of 42°. What is its apex angle?
Compare with the explanation
96°
Subtract both base angles from 180°: 180−42−42=96.
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