Geometry and Geometric Transformations
This domain checks plane-figure properties, the metric and similarity relations that let you calculate unknown lengths, areas and volumes, the effect of transformations, and analytic geometry in the Cartesian plane.
Plane figures and metric relations
- Perimeter and area of triangles, quadrilaterals, regular polygons and composite figures (decompose, then add or subtract).
- Right triangle: the Pythagorean theorem; the altitude to the hypotenuse gives h² = m·n, and each leg is the geometric mean of the hypotenuse and its projection: a² = c·m, b² = c·n. Also h = (a·b)/c.
- Similar figures: corresponding angles are equal and corresponding sides are proportional; congruence is similarity with ratio k = 1.
Transformations and their effect on measurements
Isometries - translation, rotation and reflection - preserve lengths, angles and areas. A homothety (dilation) with centre O and ratio k multiplies lengths by |k|, areas by k² and volumes by k³; a negative k places the image on the opposite side of O.
Analytic geometry
Distance between two points, midpoint coordinates, slope m = (y₂ − y₁)/(x₂ − x₁). Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1. Be able to write a line's equation in slope-intercept, general or symmetric form.
Worked example
A rectangular photo measures 10 cm by 15 cm. It is enlarged so that its area becomes 1350 cm². What are the new dimensions? Original area: 10 × 15 = 150 cm². Area ratio: 1350 ÷ 150 = 9. Since areas scale by k², k² = 9 and k = 3 (positive, because it is an enlargement). New dimensions: 10 × 3 = 30 cm and 15 × 3 = 45 cm. Check: 30 × 45 = 1350 ✓.
Traps that cost marks
- Applying k to an area instead of k², or k³ to a length.
- Using the Pythagorean theorem on a triangle that is not right-angled.
- Writing the perpendicular slope as −m rather than −1/m.
- Losing marks for missing units, or for rounding before the final step.