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?

1 Answer
Jan 31, 2018

color(green)(V = 16.5781) cubic units

Explanation:

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Triangle Area A = (1/2) * a * h

BC = a = sqrt((5-3)^2 + (9-1)^2) = 8.2462

Slope of BC m_a = (9-1) / (5-3) = 4

Equation of BC

(y - 1) / (9 - 1) = (x - 3) / (5 - 3)

2y - 2 = 8x - 24

y - 4x = -11 Eqn (1)

Slope of altitude through A is m__h = -1/4

Equation of altitude h is

y - 7 = -(1/4) (x - 6)

4y + 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 AD = h = sqrt((6-(78/17))^2 + (7 - (128/17))^2) = 1.5078

Area of Triangle color(green)A = (1/2) b h = (1/2) * 8.2462 * 1.5078 = color(green)(6.2168)

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

color(green)(V = (1/3) 6.2168 * 8 = 16.5781)