Two scratches on a round metal tray each measure 12 cm end to end. The tray has radius 10 cm and centre O. Find how far each is from O. (2)
Question 1 Report
Two scratches on a round metal tray each measure 12 cm end to end. The tray has radius 10 cm and centre O.
- Find how far each is from O. (2)
Answer Details
Each scratch is a chord of the circle. The perpendicular from the centre to a chord bisects it, which creates a right-angled triangle whose hypotenuse is a radius.
- Half of each 12 cm chord is 6 cm, and the radius to an end of the chord is 10 cm, so if \(d\) is the distance from \(O\) to the chord, \(d^2 = 10^2 - 6^2 = 100 - 36 = 64\). [1]
- \(d = 8\) cm, and since both scratches have the same length, both lie 8 cm from \(O\). [1]
This is the general rule that chords of equal length in one circle are equidistant from the centre, and it applies whatever direction the two scratches run in.
The 10 cm radius is the hypotenuse, so it must be squared and the half-chord subtracted from it; halving the chord before using Pythagoras is the step most often missed, and using 12 cm would give no real answer at all.
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