I'm Up and Down, and Round and Round

Chords, their distance from the centre, and the angles they make at the centre and on the circle, including cyclic quadrilaterals.

Key ideas

The perpendicular from the centre to a chord bisects the chord; the line from the centre to the midpoint of a chord is perpendicular to it.
Equal chords are equally far from the centre, and chords equally far from the centre are equal.
Equal chords make equal angles at the centre.
The angle a chord makes at the centre is twice the angle it makes at any point on the remaining part of the circle.
Angles in the same segment are equal, and the angle in a semicircle is 90°.
In a cyclic quadrilateral opposite angles add up to 180°; if they do, the quadrilateral is cyclic.
An exterior angle of a cyclic quadrilateral equals the interior opposite angle.

Chord, distance and radius	r² = d² + (half the chord)²
Centre and circle	angle at centre = 2 × angle at the circle
Cyclic quadrilateral	∠A + ∠C = ∠B + ∠D = 180°

Using the whole chord instead of half of it in r² = d² + (c/2)².
Halving instead of doubling: the centre angle is the bigger one.
Adding adjacent angles of a cyclic quadrilateral to 180° (it is the opposite ones).

Two parallel chords of lengths 30 cm and 40 cm are drawn on opposite sides of the centre of a circle of radius 25 cm. Find the distance between them.

Working: distances from the centre: √(25² − 15²) = 20 and √(25² − 20²) = 15; sum = 35

Answer: 35 cm