Expert: Chen Min - 1/26/2011

Question
Hello Chen,

I'm hoping you can help me with this question and give me some clues as to how to solve:

A patio umbrella is being designed in the shape of a square-based pyramid.  The lengths of the sides of the base are all 2 m.  The height (AD) of the umbrella is 0.5 m.  Find the lengths of the sides of the triangular pieces (sides AB and AC) and the measure of angle BAC.

I'm having trouble with this because there are many unknowns.  Any help would be greatly appreciated.

~Patti~

Answer
Look at the base, surely the height AD extends through the center of it, and the vertically downward projection of all slanting sides like DB and DC (of the triangle pieces) will divide the base into four identical triangles.

These triangles are all 45-45-90, hence with hypotenuse being 2, their other side is sqrt2. Now look at triangles with sides consisting the height AD, the sqrt2 side DB, and the sides of triangular side pieces AB, they also form Right Angled Triangles, and the hypotenuse being the sides of triangular pieces, which is sqrt(0.5^2+2)=sqrt3.(lengths of the sides of the triangular pieces AB and AC)

So the triangular pieces have sides sqrt3, sqrt3, 2. The angle BAC can be found with cosine rule.

It might look confusing...draw a picture yourself.