Level 7: Similarity and trigonometry

Geometry · level 7 of 8 · rough age 15-16 (a guide for parents; anyone can start anywhere)

This level is about measuring what you cannot reach. It starts with similar shapes, where one triangle is a scaled copy of another, and turns that idea into trigonometry: sine, cosine and tangent, then the sine and cosine laws that solve every triangle there is. It ends with circles, first through the angle theorems that the Greeks proved, then through equations that let algebra find where circles and lines meet.

Lesson	You will be able to
1. Similar triangles and scale factors	Prove triangles similar with AA, SSS or SAS, find missing sides, and scale areas by \(k^2\) and volumes by \(k^3\)
2. Sine, cosine and tangent	Use SOH CAH TOA to find sides and angles of right triangles, and know the exact values for 30°, 45° and 60°
3. Angles of elevation and depression	Measure heights and distances you cannot reach, including with two angles from two places
4. The sine law and the ambiguous case	Prove and use the sine law, and decide whether two sides and an angle give 0, 1 or 2 triangles
5. The cosine law; solving any triangle	Prove and use the cosine law, find areas with \(\tfrac12 ab\sin C\), and choose the right tool for any triangle
6. Circle theorems	Prove and use the angle at the centre, angles in the same segment, cyclic quadrilaterals and tangent facts
7. Equations of circles	Write and read \((x-a)^2 + (y-b)^2 = r^2\), complete the square, and find where lines and circles meet

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