UKMT Maths Question - help needed to explain answer

(5 Posts)
wellsoftunbridge Sun 14-Aug-16 23:38:22

This is a JMO Mentoring scheme question, and answer:

Q: In a quadrilateral ABCD, side AB = side BC = diagonal CA = 2 and side AD = 1. D is located so that a circle using diagonal BD as a diameter will pass through A. Prove that BD = sq-root of 5.

A: Angle BAD = 90 degrees because BD is the diameter of the circle through A. Using Pythagoras's rule on triangle BAD we can work out that BD squared= 5.

I want to explain this to my son but the bit I'm struggling with is to find a simple explanation for why Angle BAD = 90 degrees. It's stated as fact in the answer, and with a bit of fiddling around we (or rather DH) did manage to prove it, but is there a named theorem for it that we should have just been able to quote?

angelcake20 Mon 15-Aug-16 00:23:23

It's called Thales' Theorem, though I don't remember it being named at school, just being taught the fact and proof.

https://en.m.wikipedia.org/wiki/Thales%27_theorem

wellsoftunbridge Mon 15-Aug-16 08:05:04

Thanks angelcake. That's perfect.
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booklooker Mon 15-Aug-16 09:40:51

Cor! I've been teaching maths for decades, and I never knew it had its own name.

I have always just stated that an angle in a semi-circle is a right angle

GHGN Wed 17-Aug-16 17:54:15

This one comes up quite a bit in Olympiad questions. Some other useful things at this level that your son should know is the concurrent property of median, altitude, angle bisector and perpendicular lines. They tend to be given but it is interesting and simple enough to know the proof.

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