# The base of a triangular pyramid is a triangle with corners at (6 ,7 ), (3 ,1 ), and (5 ,9 ). If the pyramid has a height of 8 , what is the pyramid's volume?

Jan 31, 2018

$\textcolor{g r e e n}{V = 16.5781}$ cubic units

#### Explanation:

Triangle Area $A = \left(\frac{1}{2}\right) \cdot a \cdot h$

$B C = a = \sqrt{{\left(5 - 3\right)}^{2} + {\left(9 - 1\right)}^{2}} = 8.2462$

Slope of BC ${m}_{a} = \frac{9 - 1}{5 - 3} = 4$

Equation of BC

$\frac{y - 1}{9 - 1} = \frac{x - 3}{5 - 3}$

$2 y - 2 = 8 x - 24$

$y - 4 x = - 11$ Eqn (1)

Slope of altitude through A is ${m}_{_} h = - \frac{1}{4}$

Equation of altitude h is

$y - 7 = - \left(\frac{1}{4}\right) \left(x - 6\right)$

$4 y + x = 34$ Eqn 2

Solving equations (1) & (2), we get the coordinates of D, the base of altitude h

#D (78/11, 128/17)

Height of altitude $A D = h = \sqrt{{\left(6 - \left(\frac{78}{17}\right)\right)}^{2} + {\left(7 - \left(\frac{128}{17}\right)\right)}^{2}} = 1.5078$

Area of Triangle $\textcolor{g r e e n}{A} = \left(\frac{1}{2}\right) b h = \left(\frac{1}{2}\right) \cdot 8.2462 \cdot 1.5078 = \textcolor{g r e e n}{6.2168}$

Volume of Pyramid V = (1/3) * Base Area * Height

$\textcolor{g r e e n}{V = \left(\frac{1}{3}\right) 6.2168 \cdot 8 = 16.5781}$