A circular water butt on an allotment, centre \(O\) and radius \(r\) cm, has a straight drainpipe support tangent to its edge at \(T\), meeting a fence post...
Question 1 Report
A circular water butt on an allotment, centre \(O\) and radius \(r\) cm, has a straight drainpipe support tangent to its edge at \(T\), meeting a fence post at \(P\), as shown. \(TP=24\) cm and \(OP=25\) cm.
- Give a reason why angle \(OTP=90^{\circ}\). (1)
- Work out the radius, \(r\). (2)
Answer Details
(a) A tangent to a circle always meets the radius drawn to the point of contact at a right angle, so angle \(OTP = 90^{\circ}\). [1 mark]
(b) Since angle \(OTP = 90^{\circ}\), triangle \(OTP\) is right-angled at \(T\), with \(OP\) as the hypotenuse. By Pythagoras' theorem, \(r^{2} = OP^{2}-TP^{2} = 25^{2}-24^{2} = 625-576 = 49\), so \(r = \sqrt{49} = 7\) cm. [2 marks]
The numbers \(7\), \(24\), \(25\) form a Pythagorean triple, so the radius comes out as an exact whole number.
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