The base of a triangular pyramid is a triangle with corners at $(8, 5)$ $(8 ,5 )$ , $(7, 2)$ $(7 ,2 )$ , and $(4, 6)$ $(4 ,6 )$ . If the pyramid has a height of $7$ $7$ , what is the pyramid's volume?

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The base of a triangular pyramid is a triangle with corners at $(8, 5)$ $(8 ,5 )$ , $(7, 2)$ $(7 ,2 )$ , and $(4, 6)$ $(4 ,6 )$ . If the pyramid has a height of $7$ $7$ , what is the pyramid's volume?

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Answer 1 · 2018-07-30T03:08:28

$V = \frac{1}{3} A_{b} h = (\frac{1}{3}) \cdot 7.1171 \cdot 7 = 16.6065, cubic units$ $color(green)(V = 1/3 A_b h = (1/3) * 7.1171 * 7 = 16.6065, " cubic units"$

Explanation:

Given : $A (8, 5), B (7, 2), C (4, 6)$ $A (8,5), B (7,2), C (4,6)$ , h = 7#

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

$d = \sqrt{{(x_{2} - x_{1})}_{2}^{2} + {(y_{2} - y_{1})}_{2}^{2}}$ $d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)$

$a = \sqrt{{(7 - 4)}^{2} + {(2 - 6)}^{2}} = 5$ $a = sqrt((7-4)^2+(2-6)^2) = 5$

$b = \sqrt{{(4 - 8)}^{2} + {(6 - 5)}^{2}} = \sqrt{17}$ $b = sqrt((4-8)^2+(6-5)^2) = sqrt 17$

$c = \sqrt{{(8 - 7)}^{2} + {(5 - 2)}^{2}} = \sqrt{10}$ $c = sqrt((8-7)^2+(5-2)^2) = sqrt 10$

Semi perimeter of base triangle $s = \frac{a + b + c}{2}$ $s = (a+b+c)/2$

$s = \frac{5 + \sqrt{17} + \sqrt{10}}{2} = \frac{12.2854}{2} \approx 6.2427$ $s = (5 + sqrt 17 + sqrt 10)/2 = 12.2854/2 ~~ 6.2427$

Area of triangle $A_{t} = \sqrt{s (s - a) (s - b) (s - c)}$ $A_t = sqrt(s (s-a) (s-b) (s-c))$

$A_{t} = \sqrt{6.2427 \cdot (6.2427 - 5) \cdot (6.2427 - \sqrt{17}) \cdot (6.2427 - \sqrt{10})}$ $A_t = sqrt(6.2427 * (6.2427-5)*(6.2427- sqrt 17)*(6.2427-sqrt 10))$

#A_t ~~ 7.1171

Volume of pyramid $V = (\frac{1}{3}) \cdot A_{t} \cdot h$ $V = (1/3) * A_t * h$

$V = (\frac{1}{3}) \cdot 7.1171 \cdot 7 = 16.6065, cubic units$ $color(green)(V = (1/3) * 7.1171 * 7 = 16.6065, " cubic units"$

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The base of a triangular pyramid is a triangle with corners at $(8, 5)$ $(8 ,5 )$ , $(7, 2)$ $(7 ,2 )$ , and $(4, 6)$ $(4 ,6 )$ . If the pyramid has a height of $7$ $7$ , what is the pyramid's volume?

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Explanation:

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The base of a triangular pyramid is a triangle with corners at $(8, 5)$ $(8 ,5 )$ , $(7, 2)$ $(7 ,2 )$ , and $(4, 6)$ $(4 ,6 )$ . If the pyramid has a height of $7$ $7$ , what is the pyramid's volume?