shape and space 2

Special Quadrilaterals Bookmark this page A general quad lateral has four sides and the sum of its interior angles is 360; unless told otherwise, no sides are equal and no angles are equal. However there are several special cases. SQUARE In a square, all four sides are equal and all four angles are right angles. A square has four lines of symmetry. RECTANGLE In a rectangle, the opposite sides are equal and all four angles are right angles. A rectangle has two lines of symmetry.   PARALLELOGRAM       A parallelogram has opposite sides that are equal and parallel, and opposite angles that are equal. A parallelogram has no lines of symmetry but it does have rotational symmetry of order 2. RHOMBUS In a rhombus the opposite angles are equal, the opposite sides are parallel and all four sides are the same length. A rhombus has two lines of syuunetiy (the diagonals). KITE A kite has one pair of opposite angles equal and two pairs of adjacent sides equal. A kite has one line of symmetry. It has no rotational symmetry.   TRAPEZIUM A trapezium has one pair of opposite sides parallel. Usually it has no line of symmetry. However, in the special case when the two sloping sides are the same length there is oneline of symmetry. This is called an isosceles trapezium. Its non-parallel sides are e4ual andtwo pairs of adjacent angles are equal.

Transformations A translation occurs when a shape has been moved from one place to another. It is equivalent of picking up the shape and putting it down somewhere else. Vectors are used to describe such transformations. When describing a reflection, you need to state the line which the shape has been reflected in. When describing a rotation, the centre and angle of rotation are given. If you wish to use tracing paper to help with rotations: draw the shape you wish to rotate onto the tracing paper and put this over shape. Push the end of your pencil down onto the tracing paper, where the centre of rotation is and turn the tracing paper through the appropriate angle. The resultant position of the shape on the tracing paper is where the shape is rotated to. Enlargements Enlargements have a centre of enlargement and a scale factor. 1) Draw a line from the centre of enlargement to each vertex ('corner') of the shape you wish to enlarge. Measure the lengths of each of these lines. 2) If the scale factor is 2, draw a line from the centre of enlargement, through each vertex, which is twice as long as the length you measured. If the scale factor is 3, draw lines which are three times as long. If the scale factor is 1/2, draw lines which are 1/2 as long. Example: The centre of enlargement is marked. Enlarge the triangle by a scale factor of 2.

Vectors A vector quantity has both length (magnitude) and direction. The opposite is a scalar quantity, which only has magnitude. Vectors can be denoted by AB, a, or AB (with an arrow above the letters). If a = (3) then the vector will look as follows:          (2) NB1: When writing vectors as one number above another in brackets, this is known as a column vector. NB2: in textbooks and here, vectors are indicated by bold type. However, when you write them, you need to put a line underneath the vector to indicate it. Multiplication by a Scalar When multiplying a vector by a scalar (i.e. a number), multiply each component of the vector by the scalar. Example: If a = ( 3 ), and b = 2a, sketch a and b.          ( 2 ) If a = ( 3 ),   2a = ( 6 )          ( 2 )           ( 4 )   Vector Manipulation   Example: If a = (-5) and b = ( 2), find the magnitude of their resultant.          ( 3)            ( 1) The resultant of two or more vectors is their sum. The resultant therefore is (-3).                                      ( 4) The magnitude of this is Ö(-3² + 4²) = Ö(9 + 16) = Ö(25) = 5 The addition and subtraction of vectors can be shown diagrammatically. To find a + b, draw a and then draw b at the end of a. The resultant is the line between the start of a and the end of b. To find a - b, find -b (see above) and add this to a. Example: Unit Vectors A unit vector has a magnitude of 1. The unit vector in the direction of the x-axis is i and the unit vector in the direction of the y-axis is j. For example on a graph, 3i + 4j would be at (3 , 4). This method is another method of writing down vectors. Example: 3i + j  plus  5i - 4j =   8i - 3j. This is equivalent to: ( 3) + ( 5 )  =  ( 8 ) ( 1 )    ( -4)      ( -3)

special quadeilaterals

transformation

vectors