### Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

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?

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.

# Incentre Angle

##### Age 14 to 16 Short Challenge Level:

Let
$\angle OLM = \angle OLN = a^{\circ},$
$\angle OML = \angle OMN = b^{\circ}$  and
$\angle LOM = c^{\circ}$

Angles in a triangle add up to $180^{\circ}$, so from $\triangle LMN$, $$2a^{\circ}+2b^{\circ}+68^{\circ} = 180^{\circ}$$ which gives $$2(a^{\circ}+b^{\circ})=112^{\circ}$$ In other words $$a^{\circ}+b^{\circ}=56^{\circ}$$

Also, from $\triangle LOM$, $$a^{\circ}+b^{\circ}+c^{\circ}=180^{\circ}$$ and so
\eqalign{ c^{\circ}&= 180^{\circ} - (a^{\circ}+b^{\circ})\cr &= 180^{\circ}-56^{\circ}\cr &=124^{\circ}}

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.