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Look Before You Leap

Can you spot a cunning way to work out the missing length?

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Triangle Midpoints

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

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Trapezium Four

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Two Right Angles

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Triangles $QPR$ and $RPS$ are similar. So:$\frac{PR}{PS}=\frac{PQ}{PR}$. Hence $PR^2=PQ\times PS=\frac{7}{3}\times\frac{48}{7}=16$.
So $PR$ is $4$ units long.

This problem is taken from the UKMT Mathematical Challenges.