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

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

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Answer 1 · 2018-01-05T19:28:14

$4$ $4$

Explanation:

The area of a triangle with vertices $(x_{1}, y_{1})$ $(x_1, y_1)$ , $(x_{2}, y_{2})$ $(x_2, y_2)$ , $(x_{3}, y_{3})$ $(x_3, y_3)$ is given by the formula:

$A = \frac{1}{2} | x_{1} y_{2} + x_{2} y_{3} + x_{3} y_{1} - x_{1} y_{3} - x_{2} y_{1} - x_{3} y_{2} |$ $A = 1/2 abs(x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2)$

Letting $(x_{1}, y_{1}) = (6, 7)$ $(x_1, y_1) = (6, 7)$ , $(x_{2}, y_{2}) = (4, 5)$ $(x_2, y_2) = (4, 5)$ and $(x_{3}, y_{3}) = (8, 7)$ $(x_3, y_3) = (8, 7)$ we find that the area of the base of the given pyramid is:

$\frac{1}{2} | 6 \cdot 5 + 4 \cdot 7 + 8 \cdot 7 - 6 \cdot 7 - 4 \cdot 7 - 8 \cdot 5 |$ $1/2abs(color(blue)(6) * color(blue)(5)+color(blue)(4) * color(blue)(7) + color(blue)(8) * color(blue)(7) - color(blue)(6) * color(blue)(7) - color(blue)(4) * color(blue)(7) - color(blue)(8) * color(blue)(5))$

$= \frac{1}{2} | 30 + 28 + 56 - 42 - 28 - 40 | = 2$ $=1/2abs(30+28+56-42-28-40) = 2$

Another way of seeing this is by considering the points $(6, 7)$ $(6, 7)$ and $(8, 7)$ $(8, 7)$ as the base of the triangle, which is of length $2$ $2$ . Then the apex of the triangle at $(4, 5)$ $(4, 5)$ is at height $2$ $2$ above the base. Then the area of the triangle is:

$\frac{1}{2} \cdot base \cdot height = \frac{1}{2} \cdot 2 \cdot 2 = 2$ $1/2 * "base" * "height" = 1/2 * color(blue)(2) * color(blue)(2) = 2$

Then the volume of a pyramid is:

$\frac{1}{3} \cdot base \cdot height = \frac{1}{3} \cdot 2 \cdot 6 = 4$ $1/3 * "base" * "height" = 1/3 * color(blue)(2) * color(blue)(6) = 4$

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

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The base of a triangular pyramid is a triangle with corners at (6,7)(6 ,7 ), (4,5)(4 ,5 ), and (8,7)(8 ,7 ). If the pyramid has a height of 66 , what is the pyramid's volume?

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