### Pent

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

### Pentakite

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

### Golden Mathematics

A voyage of discovery through a sequence of challenges exploring properties of the Golden Ratio and Fibonacci numbers.

# Pentabuild

##### Stage: 5 Challenge Level:

 The film strip shows the steps in the construction of the regular pentagon. Copy this straight edge and compass construction for yourself and explain why it produces a regular pentagon. The description of the construction below, and the information in the notes, should help you to explain the construction.
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Here are the steps shown in the film sequence:

1. Draw a circle $C_1$ centre $O$ diameter $PQ$.

The circle $C_1$ has radius 1 unit; what is its equation?

2. Draw the perpendicular bisector of $PQ$ cutting $PQ$ at $O$ and $C_1$ at $A$ and $Y$.

3. Draw perpendicular bisectors of $PO$ and $OQ$ cutting $PQ$ at $R$ and $S$.

Find the length $YS$

4. Draw circles $C_2$ and $C_3$ centres $R$ and $S$ and radii $RO$ and $SO$.

5. Join $R$ and $S$ to the point $Y$ cutting $C_2$ at $T$ and $U$ and $C_3$ at $V$ and $W$.

6. Draw circle $C_4$ centre $Y$ radius $YW=YU$ cutting $C_1$ at $D$ and $C$.

What is the equation of $C_4$? Find the value of $y$ at the intersection of $C_1$ and $C_4$ .

7. Draw circle $C_5$ centre $Y$ radius $YT=YV$ cutting $C_1$ at $E$ and $B$.

What is the equation of $C_5$ ?

Find the value of $y$ at the intersection of $C_1$ and $C_5$.

At $B$ and $E$ $x^2 + y^2 +2y +1 = 2y + 2 = (3 + \sqrt 5)/2$ so

8. Join $AB$, $BC$, $CD$, $DE$, $EA$.

How would you adapt this construction to produce a regular decagon?