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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.

Linkage

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?

Shared Vertex

Age 11 to 14 Short Challenge Level:


Triangle $BCD$ is isosceles, as $BC=BD$. Therefore, $\angle BCD = \angle BDC = 65^\circ$.







Angles in a triangle add up to $180^\circ$, so applying this in $BCD$ gives:
$\angle CBD = 180^\circ - \angle BCD - \angle BDC = 180^\circ - 65^\circ - 65^\circ = 50^\circ$.

$\angle CBD$ and $\angle ABE$ are opposite angles at $B$, so are equal. Therefore $\angle ABE = 50^\circ$.

Since the angles in triangle $ABE$ add up to $180^\circ$, $x = 180^\circ - \angle BEA - \angle ABE = 180^\circ - 90^\circ - 50^\circ = 40^\circ$.

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