Q21. - Geometry

Question

Find the following for the regular triangular pyramid.

If AM = 9 and VA = 10, find h and l.

Step-by-Step Solution

Answer

$h=8\text{ and }l=\sqrt{73}$

Step 1. Given information.

The figure provided in the question,

$AM=9\text{ and }VA=10$

Step 2. Determine l .

Since$AM=9$ , 

$\begin{array}{c}\frac{\sqrt{3}}{2}a=9\\ a=\frac{18}{\sqrt{3}}\text{ [multiply]}\\ a=6\sqrt{3}\text{ [simplify]}\end{array}$

Now,$VA=10$ ,

$\begin{array}{c}l=\sqrt{{\left(10\right)}^2-{\left(\frac{6\sqrt{3}}{2}\right)}^2}\\ =\sqrt{100-27}\text{ [square]}\\ =\sqrt{73}\text{ [subtract]}\end{array}$

Step 3. Determine h .

Now determine the height of pyramid.

$\begin{array}{c}{h}^2+{\left(3\right)}^2={\left(\sqrt{73}\right)}^2\\ h^2=73-9\text{ [sqaure and subtract]}\\ h^2=64\text{ [subtract]}\\ h=8\text{ [square root]}\end{array}$