### 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?

### 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.

### Fitting In

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

# The Eyeball Theorem

### Why do this problem?

In this problem, the dynamc geometry applet provides an opportunity for experimenting and making conjectures. The proof can be done entirely by similar triangles but there are several possible methods. A useful problem-solving skill to apply is to simplify the image and another is the magic of the key construction line that opens up new possibiliites.

### Possible approach

Give the class time to experiment and make their own conjectures.

Make a list and chose one as a focus. The 'obvious' one is that the two chords are equal but they might, for example, notice that they are perpendicular to the line joining the centres.

Having identified a conjecture of interest, ask learners to write on separate cards statements about the figure that they know are true or think might be true (and distinguish between them).

In groups look at the the statements they have and try to arrange them into those they think are useful and those not and those aperson can justufy and those they cannot.
Now try to order or arrange them further discussing the possible use they might be or insights they might offer.
Can they come up with any further statements?

Share ideas ready to put together a more formal proof. Why not emphasisethe messiness of getting to a stage where enough is known, and a direction has emerged, before formalising a proof.

### Key questions

• Can you see any similar triangles?
• Can you see symmetry in the diagram?
• Is there a line you could draw that might give us a new insight?
• Do we need all the diagram?

### Possible extension

Try the problem Belt

### Possible support

Encourage the learners to draw the dynamic diagram for themselves using Geogebra.