Geometry
Shapes, graphs and trigonometry — constructing figures, plotting graphs, and calculating angles and coordinates.
Geometry — summary notes
- Trigonometry relates the angles of a right-angled triangle to the ratios of its sides — remembered as SOH-CAH-TOA.
- Pythagoras' theorem links the three sides of a right-angled triangle and is used constantly for resultant forces and diagonal distances.
- Graphs of equations and functions show the relationship between variables; the gradient of a straight line is its rate of change.
- Rectangular (x, y) and polar (r, θ) coordinates describe the same point two ways. Polar form is natural for anything expressed as a magnitude and a direction — a force, a wind vector, an AC phasor.
- Angles convert between degrees and radians; radians are the natural unit for rotational and phase calculations.
- Trigonometric ratios
- sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
- Pythagoras
- a² + b² = c² (c = hypotenuse)
- Rectangular → polar
- r = √(x² + y²), θ = tan⁻¹(y / x)
- Polar → rectangular
- x = r cos θ, y = r sin θ
- Degrees ↔ radians
- radians = degrees × π/180
- Circle
- circumference = 2πr, area = πr²
A right-angled triangle has legs of 6 cm and 8 cm. Find the hypotenuse.
c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.
A 5 m ladder leans against a wall at 60° to the ground. How far up the wall does it reach?
The height is the opposite side: sin 60° = height / 5, so height = 5 × 0.866 = 4.33 m.
Convert the rectangular coordinate (6, 8) to polar form.
r = √(6² + 8²) = √100 = 10. θ = tan⁻¹(8/6) = tan⁻¹(1.333) = 53.13°. So (10, 53.13°).
Geometry concept map
Geometry
Geometry quiz
Geometry
1. The interior angles of any triangle add up to:
2. Angles on a straight line add up to:
3. Angles around a point add up to:
4. A right angle measures:
5. Each interior angle of an equilateral triangle is:
6. The interior angles of a quadrilateral add up to:
7. An angle bisector divides an angle into:
8. The longest side of a right-angled triangle is the:
9. On a graph, the gradient represents:
10. The general equation of a straight line is:
11. In y = 3x + 2, the gradient is:
12. In y = 3x + 2, the y-intercept is:
13. A horizontal line has a gradient of:
14. On a graph, the x-axis is the:
15. Rectangular coordinates are written as: