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.
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative


Image
Golden Triangle

The three triangles $ABC$, $CBD$ and $ABD$ are all isosceles. Find the angles in the triangles.

The sides $AB$ and $BC$ have lengths $p$ and $q$ respectively. Prove that the ratio $p/q$ is equal to the golden ratio $\frac{1}{2} (\sqrt{5}\ +1) $.

and find the ratio $q/p$.

The area of triangle $ABC$ is 2 square units. Find the areas of $CBD$ and $ABD$ exactly (i.e. find the areas in the form \[ a + b \sqrt{5} \]

where $a$ and $b$ are rational numbers)