Similarity and Symmetry — Solutions

Full worked solutions, with method and final answer for every part.

Question 1

(a)Area scale factor = (length scale factor)². Volume scale factor = (length scale factor)³.

Area SF = 1.5² = 2.25, Volume SF = 1.5³ = 3.375

Question 2

(a)Multiply the small area by the area scale factor (1.5² = 2.25).

12 × 2.25 = 27 cm²

Question 3

(a)Area scale factor = (length scale factor)².

Question 4

(a)Multiply the small area by the area scale factor (2²=4).

15 × 4 = 60 cm²

Question 5

(a)Volume scale factor is (length ratio)³, so take the cube root of each: ∛8=2, ∛27=3.

Length ratio = 2:3

Question 6

(a)An isosceles (non-equilateral) triangle has exactly one line of symmetry, through its apex.

Question 7

(a)Area scale factor = (length SF)² = 9; volume scale factor = (length SF)³ = 27.

Surface area SF = 9, volume SF = 27

Question 8

(a)Small area = 4×6=24 cm². Large rectangle is 8×12=96 cm². Ratio = 96/24.

96 ÷ 24 = 4, confirming the area scale factor

Question 9

(a)The object and its shadow form a triangle similar to the triangle formed by a known-height object (e.g. a metre stick) and its shadow at the same time, so corresponding sides are in the same ratio, letting you find the unknown height.

The ratio of height to shadow length is the same for both objects, since the triangles are similar

Question 10

(a)A square has one line of symmetry through each pair of opposite sides, plus one through each diagonal.

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