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4.11 Similarity

Syllabus
2017
Topic
4.11
Level
Higher

Learning objectives

4.11A Geometrical properties of similar figuresunderstand and use the geometrical properties that similar figures have corresponding lengths in the same ratio but corresponding angles remain unchanged4.11B Maps and scale drawingsUse and interpret maps and scale drawings.4.11HA Area scale factorsunderstand that areas of similar figures are in the ratio of the square of corresponding sides4.11HB Volume scale factorsunderstand that volumes of similar figures are in the ratio of the cube of corresponding sides4.11HC Areas and volumes of similar figuresUse the areas and volumes of similar figures to solve problems.

Use corresponding features of similar figures

Similar figures have equal corresponding angles and all corresponding lengths in one constant ratio. The orientation may change, so correspondence must be established before calculating.

Step Action
1 match equal angles or distinctive vertices
2 write corresponding sides in the same order
3 find linear scale factor k=neworiginalk=\frac{\text{new}}{\text{original}}
4 multiply every original length by kk

A valid scale factor gives the same ratio for every corresponding side. Angles remain unchanged and are never multiplied by kk.

Do not pair sides merely because they occupy the same place on the page; rotated or reflected similar figures can reverse the visual order.

Use maps and scale drawings

A scale links a measured drawing length to a real length. For '1 cm represents 80 km', a drawing measurement of dd cm represents 80d80d km.

Direction Operation
drawing to real measure, then multiply by scale value
real to drawing convert units, then divide by scale value
ratio scale 1:n1:n 1 drawing unit equals nn real units

Measure between the specified points with the same ruler convention used by the scale, keep units explicit, and allow for stated measurement tolerance before comparing routes or distances.

A straight-line map distance is not automatically the distance travelled along roads, and centimetres cannot be combined directly with kilometres.

Use area scale factors

If corresponding lengths have scale factor kk, corresponding areas have scale factor k2k^2. Conversely, an area scale factor aa gives linear scale factor a\sqrt a.

Known Required factor
length factor kk area factor k2k^2
area factor aa length factor a\sqrt a
area ratio A2:A1A_2:A_1 length ratio A2/A1\sqrt{A_2/A_1}

Keep the direction consistent—new divided by original—then apply the squared factor to every corresponding area, including curved surface area.

Doubling every length makes area four times as large, not twice as large; area is two-dimensional.

Use volume scale factors

If corresponding lengths have scale factor kk, corresponding volumes have scale factor k3k^3. Conversely, a volume scale factor vv gives linear scale factor v3\sqrt[3]{v}.

Known Required factor
length factor kk volume factor k3k^3
volume factor vv length factor v3\sqrt[3]{v}
volume ratio V2:V1V_2:V_1 length ratio V2/V13\sqrt[3]{V_2/V_1}

Match corresponding solids, write the factor direction, cube only the linear factor, and attach cubic units to the result.

A volume ratio is not squared: a solid has three scaled dimensions, so the linear factor is cubed.

Connect lengths, areas and volumes of similar figures

For the same pair of similar figures, one linear factor kk controls all measures: lengths scale by kk, areas by k2k^2, and volumes by k3k^3.

From To Operation
area factor length factor square root
volume factor length factor cube root
area factor volume factor take square root, then cube
volume factor area factor take cube root, then square

Choose one direction and convert the given ratio back to the linear factor before moving to the required dimension. Apply totals or differences only after corresponding measures have been expressed consistently.

Do not apply an area ratio directly to a volume or vice versa; both must pass through the common linear scale factor.