Geometry and measures, Year 9: congruence, similarity and transformations

Congruent shapes

HintOne would fit exactly on top of the other, even if it had to be turned or flipped first.

The whyTurning a shape round or flipping it over does not stop it being congruent to the original. What must match is every side length and every angle.

SAS

HintNotice where the 40° corner sits in relation to the two measured lengths.

The whyAngle ABC is at B, between sides AB and BC, so each triangle is fixed by two sides and the angle between them: side, angle, side. The order of the letters matters, because the angle must be the one trapped between the two sides.

They can be different sizes

HintThink of a photograph and a poster-sized print of it.

The whyThree matching angles fix the shape of a triangle and nothing more: one can be an enlargement of the other. Every congruence condition therefore includes at least one side.

RHS

HintIts three letters stand for the square corner, the longest edge and one more edge.

The whyRHS means right angle, hypotenuse, side. It joins SSS, SAS and ASA as the four conditions that each guarantee two triangles are congruent.

9 cm

HintSwap each letter of the side you want for its partner in the first triangle.

The whyE matches B and F matches C, so EF matches BC. Once two triangles are known to be congruent, every side and angle of one equals its partner in the other, which is how congruence is used to justify equal lengths.

Similar shapes

HintOne is an enlargement of the other.

The whyIn everyday speech the word means "alike". In mathematics it is exact: every angle unchanged, every length multiplied by the same number.

15 cm

HintUse the pair of widths to find the number the small rectangle has been multiplied by.

The whyThe scale factor is 10 ÷ 4 = 2.5, found from a pair of matching sides. Every length is multiplied by it, so the larger length is 6 × 2.5 = 15.

No — the short sides are multiplied by 3 but the long sides by 4

HintWork out the multiplier for each pair of matching edges separately.

The whyFor similar shapes every pair of matching lengths gives the same scale factor. Here 6 ÷ 2 = 3 but 12 ÷ 3 = 4, so B is a different shape from A, not an enlargement of it.

They must also be the same size

HintOne of the two words allows a change of scale; the other does not.

The whySimilar shapes may be enlarged, so their lengths can differ. Congruent shapes are similar shapes with a scale factor of exactly 1: every length matches as well as every angle.

8

HintCompare the two shadows to find how many times bigger the second triangle is.

The whyThe shadows are matching sides, so the scale factor is 12 ÷ 3 = 4 and the tree is 2 × 4 metres tall. The Sun strikes both at the same angle, which is what makes the triangles similar.

A translation

HintThe shape is not turned, flipped or resized — only moved.

The whyA translated shape is congruent to the original and faces the same way. To describe one fully, say how far it moves across and how far it moves up or down.

The centre of rotation

HintThe shape has turned about some fixed point — say where.

The whyA rotation needs three things: the angle, the direction and the point it turns about. Turning 90° clockwise about (0, 0) and about (3, 1) send the same shape to different places.

A reflection in the y-axis

HintCompare each pair of matching corners: one coordinate keeps its value while the other only changes sign.

The whyEach point stays at the same height but swaps to the opposite side of the vertical axis, the same distance from it. That is a mirror image, and the mirror is the line x = 0.

5 right and 2 down

HintCompare the x-values first, then the y-values, and say which way each has moved.

The whyThe x-coordinate rises from 2 to 7 and the y-coordinate falls from 3 to 1. Every other point of the shape makes exactly the same move, so one vertex is enough to describe the whole translation.

180

HintEvery vertex ends up directly across the origin from where it started.

The whyWhen both coordinates of every point change sign, each point has travelled half a turn about (0, 0). For a half turn there is no need to say clockwise or anticlockwise, because both lead to the same place.