The base of a triangular pyramid is a triangle with corners at $(4, 2)$ $(4 ,2 )$ , $(3, 6)$ $(3 ,6 )$ , and $(7, 5)$ $(7 ,5 )$ . If the pyramid has a height of $8$ $8$ , what is the pyramid's volume?

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The base of a triangular pyramid is a triangle with corners at $(4, 2)$ $(4 ,2 )$ , $(3, 6)$ $(3 ,6 )$ , and $(7, 5)$ $(7 ,5 )$ . If the pyramid has a height of $8$ $8$ , what is the pyramid's volume?

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Answer 1 · 2017-10-18T19:11:59

$20$ $20$

Explanation:

The triangle with vertices $(4, 2)$ $(4, 2)$ , $(3, 6)$ $(3, 6)$ and $(7, 5)$ $(7, 5)$ can be drawn inside a $4 \times 4$ $4 xx 4$ square with vertices $(3, 2)$ $(3, 2)$ , $(7, 2)$ $(7, 2)$ , $(7, 6)$ $(7, 6)$ and $(3, 6)$ $(3, 6)$ , dividing it into $4$ $4$ triangles...

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The two smaller triangles are right angled triangles with legs of lengths $1$ $1$ and $4$ $4$ , so their total area is $4$ $4$ .

The larger isosceles right angled triangle has area $\frac{1}{2} \cdot 3 \cdot 3 = \frac{9}{2}$ $1/2 * 3 * 3 = 9/2$

So the total area of the given triangle is:

$4^{2} - 4 - \frac{9}{2} = \frac{15}{2}$ $4^2 - 4 - 9/2 = 15/2$

The pyramid has volume:

$\frac{1}{3} \cdot base \cdot height = \frac{1}{3} \cdot \frac{15}{2} \cdot 8 = 20$ $1/3 * "base" * "height" = 1/3 * 15/2 * 8 = 20$

Answer 2 · 2017-10-18T20:38:01

$20$ $20$

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:

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

(see https://socratic.org/s/aK6h7PjW)

So putting:

${ ((x_1, y_1) = (4, 2)), ((x_2, y_2) = (3, 6)), ((x_3, y_3) = (7, 5)) :}$

we find that the base of our pyramid has area:

$1/2 abs((4)(6)+(3)(5)+(7)(2)-(4)(5)-(3)(2)-(7)(6))$

$=1/2 abs(24+15+14-20-6-42)$

$=1/2 abs(-15)$

$=15/2$

Then the volume of the pyramid is:

$1/3 xx "base" xx "height" = 1/3 * 15/2 * 8 = 20$

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The base of a triangular pyramid is a triangle with corners at $(4, 2)$ $(4 ,2 )$ , $(3, 6)$ $(3 ,6 )$ , and $(7, 5)$ $(7 ,5 )$ . If the pyramid has a height of $8$ $8$ , 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 (4 ,2 )(4,2)(4 ,2 ), (3 ,6 )(3,6)(3 ,6 ), and (7 ,5 )(7,5)(7 ,5 ). If the pyramid has a height of 8 88 , what is the pyramid's volume?

2 Answers

Explanation:

Explanation:

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The base of a triangular pyramid is a triangle with corners at $(4, 2)$ $(4 ,2 )$ , $(3, 6)$ $(3 ,6 )$ , and $(7, 5)$ $(7 ,5 )$ . If the pyramid has a height of $8$ $8$ , what is the pyramid's volume?