Pentakite

Why do this problem?

This problem asks students to use their knowledge of regular polygons, congruent and similar triangles, and the quadratic formula to calculate the distance between two non-adjacent vertices of a pentagon. 

It could be used as an introduction to the golden ratio, or just as an interesting geometry problem on its own!

Possible approach

Show students the initial diagram.  Ask them if there are any angles they can find (and ask them how they can find them!).

Draw in the triangle $DAC$ - ask if there are any more angles they can now find.

Ask students how many triangles there are now in their diagram - can they identify which are congruent to each other?

Key questions

What do you know about the angles of a reguar pentagon?

How many triangles can you spot? What sort of triangles are they?

Can you spot any similar triangles? ... any congruent triangles?

What other lengths are there which are equal to $DA$?

What do you know about similar triangles?

Possible Extensions

Pupils could be asked to calculate the ratio of successive terms of the Fibonacci sequence, i.e. calculate

$$\frac 1 1, \frac 2 1, \frac 3 2, \frac 5 3, \frac 8 5 \cdots $$

They can then compare their answers to the value of $DA$.

As an added challenge, they could simplify the fractions:

$$\frac 1 {1+1}, \frac 1 {1 + \frac 1 {1+1}}, \frac 1 {1+\frac 1 {1+\frac 1 {1+1}}}, \frac 1 {1+\frac 1 {1+\frac 1 {1+\frac 1 {1+1}}}}, \cdots$$

There are lots of problems on Nrich based on the Golden Ratio - try searching for "Golden" in the search box!