Review question

# Can we find the lengths and angles in this pyramid? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6589

## Suggestion

An equilateral triangle $ABC$ has side length $6$ units. The three altitudes of the triangle meet at $N$. Show that $AN=2\sqrt{3}$ units.

Could we draw a picture of $ABC$ with the altitudes and $N$ marked? Can you spot any similar triangles? Or right-angled triangles?

This triangle is the base of a pyramid whose apex $V$ lies on the line through $N$ perpendicular to the plane $ABC$. Given that $VN=2$ units, prove that $\widehat{VAN}=30^\circ.$

Could we draw another picture? In three dimensions this time?

The perpendicular from $A$ to the edge $VC$ meets $CV$ produced at $R$. Prove that $AR=\frac{3}{2}\sqrt{7}$ units, and find the exact value of $\cos \widehat{ARB}$.

The phrase “$CV$ produced” means, “the line segment $CV$ extended”.

Could we draw a picture of $AVC$, marking in $R$? Which lengths in this triangle do we already know?