Difference between revisions of "2008 AMC 12A Problems/Problem 18"
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Triangle <math>ABC</math>, with sides of length <math>5</math>, <math>6</math>, and <math>7</math>, has one [[vertex]] on the positive <math>x</math>-axis, one on the positive <math>y</math>-axis, and one on the positive <math>z</math>-axis. Let <math>O</math> be the [[origin]]. What is the volume of [[tetrahedron]] <math>OABC</math>? | Triangle <math>ABC</math>, with sides of length <math>5</math>, <math>6</math>, and <math>7</math>, has one [[vertex]] on the positive <math>x</math>-axis, one on the positive <math>y</math>-axis, and one on the positive <math>z</math>-axis. Let <math>O</math> be the [[origin]]. What is the volume of [[tetrahedron]] <math>OABC</math>? | ||
− | <math>\ | + | <math>\mathrm{(A)}\ \sqrt{85}\qquad\mathrm{(B)}\ \sqrt{90}\qquad\mathrm{(C)}\ \sqrt{95}\qquad\mathrm{(D)}\ 10\qquad\mathrm{(E)}\ \sqrt{105}</math> |
==Solution== | ==Solution== |
Revision as of 01:45, 26 April 2008
Problem
Triangle , with sides of length
,
, and
, has one vertex on the positive
-axis, one on the positive
-axis, and one on the positive
-axis. Let
be the origin. What is the volume of tetrahedron
?
Solution
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Without loss of generality, let be on the
axis,
be on the
axis, and
be on the
axis, and let
have respective lenghts of 5, 6, and 7. Let
denote the lengths of segments
respectively. Then by the Pythagorean Theorem,
so
; similarly,
and
. Since
,
, and
are mutually perpendicular, the tetrahedron's volume is
which is answer choice C.
See also
2008 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |