Scale, similarity and geometric ratios explained
Ratio, proportion and rates • Ratio and proportion
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Scale factor and length ratios
Scale factor is the number that multiplies every linear measurement in a shape to produce a similar shape. If one shape is similar to another with scale factor k, corresponding lengths follow the ratio 1 : k or k : 1 depending on direction. Limiting factor: scale factor applies only to linear dimensions and only when shapes are geometrically similar (same angles and proportional sides). Cause → effect: changing the scale factor changes all corresponding lengths by the same multiplier, so doubling the scale factor doubles every length.
Scale diagrams and map scales
Scale diagrams and maps use a written scale (for example 1:50 or 1 cm = 2 m) that defines the relationship between diagram units and real units. Conversion requires consistent units: convert measurements to the same unit before applying the scale. Limiting factor: map scales assume a flat representation; distances along curved surfaces or elevation changes require additional adjustment. Cause → effect: a larger map scale (larger first number in model:real form) makes the model smaller relative to reality; applying the correct multiplier converts model measurements to real sizes and vice versa.
Similarity and proportionality of sides
Similarity means corresponding angles are equal and corresponding sides are proportional. Similar triangles provide a basis for many scale problems: if two triangles are similar, the ratio of any pair of corresponding sides is the same. Limiting factor: triangles must have matching angle measures or proportional side pairs to conclude similarity. Cause → effect: establishing similarity allows the calculation of unknown sides by applying the common side ratio (scale factor).
Trigonometric ratios linked to similarity
Trigonometric ratios (sine, cosine, tangent) express ratios of side lengths in right-angled triangles. Similar right-angled triangles have identical trigonometric ratios for matching angles, so trig values remain constant across similar triangles. Limiting factor: trigonometric ratios apply directly to right-angled triangles only; non-right triangles require alternate methods or decomposition. Cause → effect: knowing an angle and one side or a trig ratio allows determination of other sides in any triangle similar to the reference right-angled triangle by scaling the trig ratios.
Area and volume scale factors
Area scales with the square of the linear scale factor: if lengths scale by k, areas scale by k^2, expressed as ratio 1 : k^2 or k^2 : 1. Volume scales with the cube of the linear scale factor: lengths scale by k implies volumes scale by k^3. Limiting factor: area and volume rules apply only to geometrically similar shapes where proportions hold in all dimensions. Cause → effect: increasing the linear scale factor increases area more rapidly (square) and volume even more rapidly (cube); small changes in length produce larger changes in area and volume.
Key notes
Important points to keep in mind