### 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.

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?

### Estimating Angles

How good are you at estimating angles?

# Diagonal Side

##### Age 11 to 14 ShortChallenge Level
Measuring using unit squares
The side length of the square is equal to the diagonal of the unit square, as shown in the diagram on the right, where the red length is the side of the new square and the blue square is a unit square.

Sticking four of these together, as shown on the left, draws the whole square. The unit squares fit perfectly together as the red lines split them in half, and so each angle is 90$^\text o$ or 45$^\text o$.

The area of the square is shaded in the diagram on the right. It contains four half-unit-squares - so its area is four half-units, or two whole units.

Using Pythagoras' Theorem to find the side length
If this square is a unit square, then its diagonal, shown in red, will be the side length of the square.

Applying Pythagoras' Theorem to the triangle will give us information about this length, here called $c$: $1^2+1^2=c^2\Rightarrow2=c^2$

But $c^2$ is the area of the square with side length $c$. So if $c^2=2$, then the area is $2$.

You can find more short problems, arranged by curriculum topic, in our short problems collection.