Find value of x, right angled triangle in a semi-circle| EXAM Question| Leaving Cert Maths |

[Q]. The point P is on the circle 'c' with centre 'O' and diameter [MN], as shown.
the length of the radius of the circle 'c' is 2*sqrt{5} cm .
|MP| = x cm, and |PN| = 2x cm.

Find the value of x.

The diameter of the circle is twice the radius. This gives us the side length |MN| = 4*sqrt{5}.

The angle at P is a right angle because following Thales's theorem "the angle at the circumference in a semicircle is a right angle". This is shown with the aid of
an animation.

Finally we use the Pythagoras' Theorem , this gives us a quadratic equation involving x. The positive value of x is the required length in cm.

0:38 Thales's Theorem
1:56 Pythagoras' Theorem

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