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Shape and Space

Transforming shapes

Reflection: With reflection the object and its image are congruent because they are the same size and shape.

Rotation: You need three things to describe a rotation: (a) the centre (b) the angle (c) the direction (e.g. clockwise).

Enlargement: The scale factor of an enlargement can be found by dividing corresponding lengths on two pictures.

Reduction: Even though a shape has undergone a reduction, mathematicians prefer to call it an enlargement with a fractional scale factor.

Translation: A translation is simply a 'shift'. There is no turning or reflection and the object stays the same size. Translations are described by vector. In a vector the top number gives the number of units across (positive to the right) and the bottom number gives the number of units up/down (positive upwards).

Translation with a vector

Tessellations: A tessellation is formed when a shape (or shapes) fit together without gaps to cover a surface, like jigsaw puzzles to cover a plane.

Bearings: are used where there are no roads to guide the way. Ships, aircraft and mountaineers use bearings to work out where they are. Bearings are measured clockwise from North. The bearing of A from B is the direction in which you travel to get to A from B.

Locus: In mathematics, the word locus describes the position of points which obey a certain rule. The locus can be the path traced out by a moving point.

Three important loci:

(a) Circle: The locus of points are equidistant from a fixed point O, it is a circle with centre O.
(b) Perpendicular bisector: The locus of points are equidistant from two fixed points A and B. It is the perpendicular bisector of the line AB. You can use compasses to draw arcs, or use a ruler and a protractor.
(c) Angle bisector: The locus of points are equidistant from two fixed lines AB and AC. It is the line which bisects the angle BAC. You may use compasses to draw arcs or use a protractor to construct the locus.

Pythagoras' theorem:
Pythagoras (569 - 500 BC) was one of the first of the great mathematical names in Greek antiquity. He settled in southern Italy an formed a mysterious brotherhood with his students who were bound by an oath not to reveal the secrets of numbers and who exercised great influence. They laid the foundations of arithmetic through geometry and were among the first mathematicians to develop the idea of proof.



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