Mathematics·Geometry · NSSCO 5.3
Symmetry & circle geometry
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This lesson is full of neat patterns and tidy rules. We start with the balance of symmetry — lines of symmetry and the order of rotational symmetry of flat shapes. Then we explore how the circle's perfect roundness gives us its symmetry properties (equal chords, and the perpendicular from the centre) and the tangent facts. Finally we meet the circle theorems for angles — the semicircle, the angle at the centre, angles in the same segment and the cyclic quadrilateral — and use every one to find unknown angles, always with a reason.
What you'll learn in this lesson
By the end you should be able to (NSSCO Mathematics 5.3):
- Recognise line (reflective) symmetry and rotational symmetry, including the order of rotational symmetry, of two-dimensional shapes
- Use the symmetry properties of triangles, quadrilaterals and circles
- Use the symmetry properties of a circle: equal chords are equidistant from the centre, and the perpendicular from the centre bisects a chord
- Use the property that the two tangents from an external point are equal in length
- Apply the circle theorems — angle in a semicircle = 90 degrees, tangent perpendicular to radius, angle at the centre = twice the angle at the circumference, angles in the same segment are equal, and opposite angles of a cyclic quadrilateral add to 180 degrees
- Calculate unknown angles in circles, always giving the correct geometrical reason
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