Difference between revisions of "2015 AMC 10A Problems/Problem 21"
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− | ==Problem | + | {{duplicate|[[2015 AMC 12A Problems|2015 AMC 12A #16]] and [[2015 AMC 10A Problems|2015 AMC 10A #21]]}} |
+ | ==Problem== | ||
Tetrahedron <math>ABCD</math> has <math>AB=5</math>, <math>AC=3</math>, <math>BC=4</math>, <math>BD=4</math>, <math>AD=3</math>, and <math>CD=\tfrac{12}5\sqrt2</math>. What is the volume of the tetrahedron? | Tetrahedron <math>ABCD</math> has <math>AB=5</math>, <math>AC=3</math>, <math>BC=4</math>, <math>BD=4</math>, <math>AD=3</math>, and <math>CD=\tfrac{12}5\sqrt2</math>. What is the volume of the tetrahedron? | ||
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<cmath>V = \dfrac{1}{3} Bh = \dfrac{1}{3} h \cdot BE \cdot \dfrac{6\sqrt{2}}{5} = \dfrac{24}{5},</cmath> | <cmath>V = \dfrac{1}{3} Bh = \dfrac{1}{3} h \cdot BE \cdot \dfrac{6\sqrt{2}}{5} = \dfrac{24}{5},</cmath> | ||
and so our answer is <math>\textbf{(C)}</math>. | and so our answer is <math>\textbf{(C)}</math>. | ||
+ | |||
+ | == See Also == | ||
+ | {{AMC10 box|year=2015|ab=A|num-b=20|num-a=22}} | ||
+ | {{AMC12 box|year=2015|ab=A|num-b=15|num-a=17}} |
Revision as of 20:52, 4 February 2015
- The following problem is from both the 2015 AMC 12A #16 and 2015 AMC 10A #21, so both problems redirect to this page.
Problem
Tetrahedron has
,
,
,
,
, and
. What is the volume of the tetrahedron?
Solution
Let the midpoint of be
. We have
, and so by the Pythagorean Theorem
and
. Because the altitude from
of tetrahedron
passes touches plane
on
, it is also an altitude of triangle
. The area
of triangle
is, by Heron's Formula, given by
Substituting
and performing huge (but manageable) computations yield
, so
. Thus, if
is the length of the altitude from
of the tetrahedron,
. Our answer is thus
and so our answer is
.
See Also
2015 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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 10 Problems and Solutions |
2015 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 15 |
Followed by Problem 17 |
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 |