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Two Trees

Two trees 20 metres and 30 metres long, lean across a passageway between two vertical walls. They cross at a point 8 metres above the ground. What is the distance between the foot of the trees?

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Golden Triangle

Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.

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Strange Rectangle

ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.


Stage: 5 Challenge Level: Challenge Level:1

Why do this problem?

The problem calls for simple geometrical reasoning (nothing more complicated than similar triangles) and leads to an obvious generalisation. Finding the solution is the beginning of the next stage:

how can I explain my method?
is this the best method?
what if...can I generalise this result?

Key questions

What do you know about the tangent to a circle and its radius?

What do you know about the perpendicular from the centre of a circle to a chord?

Can you spot any similar triangles?

If two sides of a right-angled triangle are in the ratio 3:5 what can we say about the third side?

Possible extension

Try The Eyeball Theorem