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

Congruent shapes

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

为什么Turning 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

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

为什么Angle 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

提示Think of a photograph and a poster-sized print of it.

为什么Three 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

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

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

9 cm

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

为什么E 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

提示One is an enlargement of the other.

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

15 cm

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

为什么The 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

提示Work out the multiplier for each pair of matching edges separately.

为什么For 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

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

为什么Similar 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

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

为什么The 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

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

为什么A 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

提示The shape has turned about some fixed point — say where.

为什么A 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

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

为什么Each 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

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

为什么The 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

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

为什么When 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.