You may also like

problem icon

Three Way Split

Take any point P inside an equilateral triangle. Draw PA, PB and PC from P perpendicular to the sides of the triangle where A, B and C are points on the sides. Prove that PA + PB + PC is a constant.

problem icon

Isosceles

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

problem icon

Pareq Calc

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel lines are 1 unit and 2 units.

Weekly Problem 23 - 2008

Stage: 3 and 4 Challenge Level: Challenge Level:1

Let $O$ be the centre of the circle.
Then $\angle POR=90^{\circ}$ as the angle subtended by an arc at the centre of a circle is twice the angle subtended by that arc at a point on the circumference of the circle.
So triangle $POR$ is an isosceles right-angled triangle with $PO=RO=4cm$. Let the length of $PR$ be $x$ cm.
Then, by Pythagoras' Theorem, $x^2=4^2+4^2=2 \times 4^2$ and so $x=4\sqrt{2}$.

This problem is taken from the UKMT Mathematical Challenges.

View the previous week's solution
View the current weekly problem
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
Copyright © 1997 - 2012. University of Cambridge. All rights reserved.
NRICH is part of the family of activities in the Millennium Mathematics Project.