The base of a triangular pyramid is a triangle with corners at #(5 ,8 )#, #(6 ,7 )#, and #(2 ,3 )#. If the pyramid has a height of #2 #, what is the pyramid's volume?

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Answer 1 · 2018-05-21T08:46:48

#color(maroon)(V = (1/3) * A_t * h = 2.65# cubic units

Given : #A (5,8), B (6,7), C (2,3)#, h = 2#

Using distance formula we can calculate the lengths of sides a, b, c.

#d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)#

#a = sqrt((6-2)^2+(7-3)^2) = sqrt32 = 5.66#

#b = sqrt((5-2)^2+(8-3)^2) = sqrt34 = 5.83#

#c = sqrt((5-6)^2+(8-7)^2) = sqrt2 = 1.41#

Semi perimeter of base triangle #s = (a+b+c)/2#

#s = (5.66 + 5.83 + 1.41)/2 = 12.9/2 = 6.45#

Area of triangle #A_t = sqrt(s (s-a) (s-b) (s-c))#

#A_t = sqrt(6.45 * (6.45-5.66)*(6.45-5.83)*(6.45-1.41))#

#A_t = 3.97#

Volume of pyramid #V = (1/3) * A_t * h#

#color(maroon)(V = (1/3) * 3.97 * 2 = 2.65# cubic units