This section focuses on the practical application of two specific circle theorems: the properties of equal chords being equidistant from the centre and the perpendicular bisector of a chord passing through the centre. We'll explore these concepts through detailed examples and step-by-step mathematical solutions.

Equal Chords and Their Distance from the Centre

Equal chords in a circle are equidistant from the centre. This principle is crucial for solving various geometrical problems efficiently.

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Example 1: Distance of Equal Chords from Centre

Given two chords, AB and CD, each of length 8 cm in a circle with a radius of 5 cm, find the distance from the centre to these chords.

Solution:

1. Identify that AB and CD are equal and thus their distance from the centre is the same.

2. Draw perpendicular bisectors from the centre, O, to the midpoints of AB and CD.

3. Use Pythagoras' theorem in the right triangle formed by the radius, the half chord, and the line from the centre to the midpoint of the chord.

Calculations:

• $OB = 5$ cm (radius)
• Half of the chord length, $BM = 4$ cm
• $OM = \sqrt{OB^2 - BM^2} = \sqrt{5^2 - 4^2} = 3$ cm

Thus, the distance from the centre to each chord is 3 cm.

The Perpendicular Bisector of a Chord

The perpendicular bisector of a chord not only bisects the chord but also passes through the circle's centre, a fundamental theorem for circle geometry.

Example 2: Finding the Distance from Centre to a Chord

Given a chord XY of length 10 cm in a circle with an 8 cm radius, calculate the chord's distance from the circle's centre.

Solution:

1. Recognise that the perpendicular bisector of XY will pass through the centre, O, and bisect XY at Z.

2. Apply Pythagoras' theorem to triangle OZX, where OX is the radius and ZX is half of the chord's length.

Calculations:

• $OX = 8$ cm (radius)
• Half of the chord length, $ZX = 5$ cm
• $OZ = \sqrt{OX^2 - ZX^2} = \sqrt{8^2 - 5^2} = \sqrt{39}$ cm

Hence, the perpendicular distance from the centre to the chord XY is $\sqrt{39}$ cm, approximately 6.24 cm.