Gold Again

Age 16 to 18

Challenge Level Yellow star

Consider the rhombus as illustrated where

$x$ is an unknown length,

$AP = AD = x$ , angle

$DAP = 36$ degrees and

$P$ is a point on the diagonal

$AC$ such that

$PB = 1$ unit.

Without using a calculator, computer or tables find the exact values of

1.

$\cos36^{\circ}\cos72^{\circ}$

2.

$\cos36^{\circ} - \cos72^{\circ}.$

3. Draw these two diagrams as accurately as you can and measure the lengths

$a$ and

$b$ . What do you notice? Can you prove it? (In each diagram there are two right angled triangles).

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