In a circle with a radius of 15 cm, a chord measures 18 cm in length. Find the perpendicular distance from the center of the circle to this chord.
12 cm
A key circle property states that the perpendicular drawn from the centre of a circle to any chord bisects that chord into two equal halves.
The chord is 18 cm, so each half of the chord measures \(\dfrac{18}{2} = 9\) cm.
Join the centre to one end of the chord; this radius equals 15 cm and acts as the hypotenuse.
The radius, the half-chord and the perpendicular distance \(d\) form a right-angled triangle, with the radius opposite the right angle.
By the Pythagoras theorem: \(d = \sqrt{15^2 - 9^2} = \sqrt{225 - 81} = \sqrt{144}\).
Since \(\sqrt{144} = 12\), the perpendicular distance is 12 cm.
Key concept: radius, half-chord and centre-to-chord distance satisfy \(r^2 = d^2 + (\tfrac{\text{chord}}{2})^2\).
The perpendicular distance is 12 cm.
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