Geometry and Measurement: Shape, Level 6
AO1: Deduce and apply the angle properties related to circles.
This means students will know and apply the sum of interior angles of a triangle (180°) and the angle between a radius and tangent (90°) to deduce the angle properties related to circles.
The angle properties of circles expected are:
- Angle at centre to any chord is twice the angle at circumference.
- Angles at the circumference to any chord are equal. .
- Angle between a chord and a tangent equals the angle in the opposite segment. .
- For a triangle in a semi-circle the angle at circumference equals 90?..
- Opposite angles in a cyclic quadrilateral add to 360?. .
Students are expected to connect at least two of these properties to find unknown angles in a given problem and to communicate their reasoning, citing the angle properties used.
AO2: Recognise when shapes are similar and use proportional reasoning to find an unknown length.
This means students will know what properties of shapes are conserved as they are enlarged (or reduced) to scale. In particular this refers to angles and the ratios of side lengths within a figure and between a figure and its enlargement. They should also apply knowledge that area increases by the square of the scale factor and volume increases by the cube of the scale factor. Students should solve problems in which they find unknown lengths of shapes that are both regular and irregular. For example: Given that the ellipses are similar, find the unknown length of the major axis.
Given the rectangles are similar find the values of c and d.
AO3: Use trigonometric ratios and Pythagoras’ theorem in two and three dimensions.
This means students will , given the required measurements, be able to connect trigonometric ratios (sine, cosine or tangent) in two dimensions with the demands of three dimensional problems, usually applying Pythagoras’ theorem in two or three dimensions in doing so. The problems should involve finding either an unknown length or angle. For example, find the angle <abc and the length of the line bc.
This objective also applies to problems where points are described using co-ordinates in two dimensions (four quadrants). For example, a triangle has corners at (2,3), (1,7), and (5,5). Find the lengths of its sides.
