Review question

# Can we find the area of a pedal triangle? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8658

## Suggestion

A triangle $ABC$ has sides $BC$, $CA$, and $AB$ of lengths $a$, $b$ and $c$ respectively, and angles at $A$, $B$ and $C$ are $\alpha$, $\beta$ and $\gamma$ where $0 \leq \alpha ,\beta ,\gamma \leq \frac{1}{2}\pi$.

1. The points $P$, $Q$ and $R$ are respectively the feet of the perpendiculars from $A$ to $BC$, $B$ to $CA$, and $C$ to $AB$ as shown.

Prove that $\text{Area of PQR} = (1 - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma) \times (\text{Area of ABC}).$

2. For what triangles $ABC$, with angles $\alpha$, $\beta$, $\gamma$ as above, does the equation $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma =1$ hold?

Use this applet to explore the behaviour for different triangles $ABC$.

The blue triangle $PQR$ is called a “pedal triangle” because it is formed from the feet of perpendiculars.

Remember that in the question, the angles at $A$, $B$ and $C$ cannot be obtuse.

When does the blue triangle disappear?