Circles, Transformations and Vectors: notes

Circle Theorems

• Angle at the centre is twice the angle at the circumference on the same arc.
• Angle in a semicircle is 90 degrees.
• Angles in the same segment are equal.
• Cyclic quadrilateral: opposite angles add to 180 degrees.
• Tangent meets the radius at 90 degrees. Two tangents from one point are equal in length.
• Chord: a line from the centre that bisects a chord meets it at 90 degrees.

Transformations

• Reflection: give the mirror line, for example x = 2 or y = x.
• Rotation: give the centre, the angle and the direction (clockwise or anticlockwise).
• Translation: a slide described by a column vector.
• Enlargement: give the centre and the scale factor. A negative scale factor puts the image on the other side of the centre.
• Only enlargement changes the size of the shape.

Vectors

• Column vector: (x over y) means x across and y up. Down and left are negative.
• Adding: add the tops and the bottoms separately.
• Scalar: 3 times (2, -1) = (6, -3).
• Magnitude: the length of (x, y) is the square root of x^2 + y^2. For (3, 4) it is 5.
• AB = b - a when a and b are the position vectors of A and B.