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

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Answer 1 · 2017-01-08T08:44:51

Pyramid's volume is $56/3$

Let they be

$A(8;5);B(6;7);C(5;1);h=8$

First, you would calculate the area of the triangle ABC, by getting the lenght of a side:

$AC=sqrt((x_C-x_A)^2+(y_C-y_A)^2)=sqrt((5-8)^2+(1-5)^2)=sqrt(9+16)=5$

and then the height relative to AC, by finding the distance between the point B and the line AC:

$m_(AC)=(y_C-y_A)/(x_C-x_A)=(1-5)/(5-8)=4/3$

Then the equation of the line AC is

$y-y_A=m_(AC)(x-x_A)$

that's

$y-5=4/3(x-8)$

$y-5=4/3x-32/3$

$3y-15=4x-32$

$4x-3y-17=0$ in the form $ax+by+c=0$

Then let's calculate the distance:

$d=|ax_B+by_B+c|/sqrt(a^2+b^2)$

$d=|4*6-3*7-17|/sqrt(4^2+3^2)=|24-21-17|/sqrt25=14/5$

Now let's calculate the area of the triangle:

$Area=(side AC)*(d)*1/2=cancel5*cancel14^7/cancel5*1/cancel2=7$

The pyramid's volume is obtained by:

$V=1/3Area_(base)*h=1/3*7*8=56/3$

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