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Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Bendy Quad

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

Three Cubes

Can you work out the dimensions of the three cubes?

Arc Radius

Age 14 to 16 Short Challenge Level:
More lengths are show on the diagrams below.



Applying Pythagoras' theorem to the diagram on the right gives $$\begin{align}(2r)^2+6^2=&(r+6)^2\\
\Rightarrow4r^2+36=&r^2+12r+36\\
\Rightarrow3r^2=&12r\\
\Rightarrow3r^2-12r=&0\\
\Rightarrow3r(r-4)=&0\\
\Rightarrow r-4=&0\hspace{12mm}(r\ne0)\\
\Rightarrow r=&4\end{align}$$
You can find more short problems, arranged by curriculum topic, in our short problems collection.