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Right Time

At the time of writing the hour and minute hands of my clock are at right angles. How long will it be before they are at right angles again?

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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.

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Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

Tent Poles

Age 11 to 14 Short Challenge Level:

As $QS=SR$, triangle $SQR$ is isosceles, so $\angle SRQ= \angle SQR =x^{\circ}$.
So by the exterior angle theorem $\angle QST=2x^{\circ}$.

Also, $\angle  TQS = 2x^{\circ}$ since $QT=TS$.

As $PT=QT$, $\angle TPQ=\angle TQP =20^{\circ}$.

Since the interior angles of triangle $PQR$ must sum to $180^{\circ}$ we obtain $$20+(20+2x+x)+x=180$$ $$40+4x=180$$ $$4x=140$$ $$x=35.$$
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.