The base of a triangular pyramid is a triangle with corners at $(2, 1)$ $(2 ,1 )$ , $(5, 2)$ $(5 ,2 )$ , and $(8, 7)$ $(8 ,7 )$ . If the pyramid has a height of $18$ $18$ , what is the pyramid's volume?

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The base of a triangular pyramid is a triangle with corners at $(2, 1)$ $(2 ,1 )$ , $(5, 2)$ $(5 ,2 )$ , and $(8, 7)$ $(8 ,7 )$ . If the pyramid has a height of $18$ $18$ , what is the pyramid's volume?

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Answer 1 · 2017-11-17T13:56:13

$36$ $36$ units cubed

Explanation:

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Volume of a triangular pyramid is:

$V = \frac{1}{3} A h$ $V=1/3Ah$

Where $A =$ $A =$ area of base, and $h =$ $h=$ height.

Dimensions of triangle:

Let the angles be at:

$A = (2, 1)$ $A=( 2 , 1 )$

$B = (5, 2)$ $B= ( 5 , 2 )$

$C = (8, 7)$ $C= ( 8 , 7 )$

Using the distance formula:

Length $A B$ $AB$

$A B = \sqrt{{(2 - 5)}^{+} {(1 - 2)}^{2}} = \sqrt{10}$ $AB=sqrt((2-5)^+(1-2)^2)=sqrt(10)$

$B C = \sqrt{{(5 - 8)}^{2} + {(2 - 7)}^{2}} = \sqrt{34}$ $BC=sqrt((5-8)^2+(2-7)^2)=sqrt(34)$

$A C = \sqrt{{(8 - 2)}^{2} + {(7 - 1)}^{2}} = \sqrt{72} = 6 \sqrt{2}$ $AC=sqrt((8-2)^2+(7-1)^2)=sqrt(72)=6sqrt(2)$

Finding $sin (A)$ $sin(A)$ of angle $A$ $A$ using the cosine rule:

$cos (A) = \frac{b^{2} + c^{2} - a^{2}}{2 b c}$ $cos(A)=(b^2+c^2-a^2)/(2bc)$

$cos (A) = \frac{{(6 \sqrt{2})}^{2} + {(\sqrt{10})}^{2} - {(\sqrt{34})}^{2}}{2 (6 \sqrt{2}) (\sqrt{10})}$ $cos(A)=((6sqrt(2))^2+(sqrt(10))^2-(sqrt(34))^2)/(2(6sqrt(2))(sqrt(10)))$

$\to = \frac{72 + 10 - 34}{24 \sqrt{5}} = \frac{48}{24 \sqrt{5}} = \frac{2}{\sqrt{5}}$ $->=(72+10-34)/(24sqrt(5))=48/(24sqrt(5))=2/sqrt(5)$

$sin (A) = sin ({cos}^{- 1} (\frac{2}{\sqrt{5}})) = \frac{\sqrt{5}}{5}$ $sin(A)=sin(cos^-1(2/sqrt(5)))=sqrt(5)/5$

Area of triangle from diagram:

$\frac{1}{2} (\sqrt{10}) sin (A) b$ $1/2(sqrt(10))sin(A)b$

$A r e a = \frac{1}{2} (\sqrt{10}) (\frac{\sqrt{5}}{5}) (6 \sqrt{2}) = \frac{60}{10} = 6$ $Area=1/2(sqrt(10))(sqrt(5)/5)(6sqrt(2))=60/10=6$

Area of pyramid:

$\frac{1}{3} (6) (18) = 3688$ $1/3(6)(18)=36color(white)(88)$

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The base of a triangular pyramid is a triangle with corners at $(2, 1)$ $(2 ,1 )$ , $(5, 2)$ $(5 ,2 )$ , and $(8, 7)$ $(8 ,7 )$ . If the pyramid has a height of $18$ $18$ , what is the pyramid's volume?

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The base of a triangular pyramid is a triangle with corners at (2,1)(2 ,1 ), (5,2)(5 ,2 ), and (8,7)(8 ,7 ). If the pyramid has a height of 1818 , what is the pyramid's volume?

1 Answer

Explanation:

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The base of a triangular pyramid is a triangle with corners at $(2, 1)$ $(2 ,1 )$ , $(5, 2)$ $(5 ,2 )$ , and $(8, 7)$ $(8 ,7 )$ . If the pyramid has a height of $18$ $18$ , what is the pyramid's volume?