Chapter 12.2, Problem 22WE

(a)

To determine

To find: The length of the sides $AM$ and $AO$ .

Answer to Problem 22WE

The length of the sides $AM$ and $AO$ is equal to $3\sqrt{3}$ and $2\sqrt{3}$ , respectively.

Explanation of Solution

Given information:

The length of the side $BC$ is equal to $6$ in the given pyramid.

  

Calculation:

The side $AM$ is the altitude of the equilateral triangle with side length equal to $6$ .

Calculate the length of the altitude $AM$ .

  $\begin{array}{c}AM=\frac{\sqrt{3}}{2}\left(BC\right)\\ =\frac{\sqrt{3}}{2}\left(6\right)\\ =3\sqrt{3}\end{array}$

The length of the side $OM$ is always the one third of the length of altitude of the triangle i.e. $AM$ .

  $\begin{array}{c}OM=\frac{1}{3}\left(AM\right)\\ =\frac{1}{3}\left(3\sqrt{3}\right)\\ =\sqrt{3}\end{array}$

The length of the side $AO$ is the subtraction of the length of side $OM$ from the length of side $AM$ .

  $\begin{array}{c}AO=AM-OM\\ =3\sqrt{3}-\sqrt{3}\\ =2\sqrt{3}\end{array}$

Therefore, the length of the sides $AM$ and $AO$ is equal to $3\sqrt{3}$ and $2\sqrt{3}$ , respectively.

(b)

To determine

To find: The value of $h$ and $l$ .

Answer to Problem 22WE

The value of $h$ and $l$ is equal to $2$ and $\sqrt{7}$ , respectively.

Explanation of Solution

Given information:

The length of the side $BC$ is equal to $6$ and the length of side $VA$ is equal to $4$ in the given pyramid.

  

Calculation:

The length of the side $MC$ is the half of the length of the side $BC$ as $M$ is the midpoint of the side $BC$ .

So, the length of the side $MC$ is equal to $3$ .

The length of the lateral edges of the pyramid is equal. So, the length of side $VC$ is equal to the length of the side $VA$ .

So, the length of the side $VC$ is equal to $4$ .

Use the Pythagoras theorem in $\Delta VMC$ for the length of side $VM$ .

  $\begin{array}{c}{\left(VC\right)}^2={\left(VM\right)}^2+{\left(MC\right)}^2\\ {\left(VM\right)}^2={\left(4\right)}^2-{\left(3\right)}^2\\ VM=\sqrt{16-9}\\ VM=\sqrt{7}\end{array}$

So, the length of $l$ is equal to $\sqrt{7}$ .

As calculated in part (a), the value of the length of the side $OM$ is equal to $\sqrt{3}$ .

Use the Pythagoras theorem in $\Delta VOM$ for the length of side $VO$ .

  $\begin{array}{c}{\left(VM\right)}^2={\left(VO\right)}^2+{\left(OM\right)}^2\\ {\left(VO\right)}^2={\left(\sqrt{7}\right)}^2-{\left(\sqrt{3}\right)}^2\\ VO=\sqrt{7-3}\\ VO=2\end{array}$

So, the length of $h$ is equal to $2$ .

Therefore, the value of $h$ and $l$ is equal to $2$ and $\sqrt{7}$ , respectively.