Difference between revisions of "2021 AMC 10A Problems/Problem 13"
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==Solution 1 (Three Right Triangles)== | ==Solution 1 (Three Right Triangles)== | ||
− | Drawing the tetrahedron out and testing side lengths, we realize that the <math>\triangle ABD, \triangle ABC,</math> and <math>\triangle ABD</math> are right triangles by the Converse of the Pythagorean Theorem. It is now easy to calculate the volume of the tetrahedron using the formula for the volume of a pyramid. If we take <math>\triangle ADC</math> as the base, then <math>AB</math> must be the height. <math>\dfrac{1}{3} \cdot \dfrac{3 \cdot 4}{2} \cdot 2</math>, so we have an answer of <math>\boxed{\textbf{(C) } 4}</math>. | + | Drawing the tetrahedron out and testing side lengths, we realize that the <math>\triangle ABD, \triangle ABC,</math> and <math>\triangle ABD</math> are right triangles by the Converse of the Pythagorean Theorem. It is now easy to calculate the volume of the tetrahedron using the formula for the volume of a pyramid. If we take <math>\triangle ADC</math> as the base, then <math>AB</math> must be the height. <math>\dfrac{1}{3} \cdot \dfrac{3 \cdot 4}{2} \cdot 2</math>, so we have an answer of <math>\boxed{\textbf{(C)} ~4}</math>. |
==Solution 2 (Bash: One Right Triangle)== | ==Solution 2 (Bash: One Right Triangle)== | ||
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&=\boxed{\textbf{(C)} ~4}. | &=\boxed{\textbf{(C)} ~4}. | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
− | |||
~MRENTHUSIASM | ~MRENTHUSIASM | ||
Revision as of 17:23, 20 June 2021
Contents
Problem
What is the volume of tetrahedron with edge lengths
,
,
,
,
, and
?
Solution 1 (Three Right Triangles)
Drawing the tetrahedron out and testing side lengths, we realize that the and
are right triangles by the Converse of the Pythagorean Theorem. It is now easy to calculate the volume of the tetrahedron using the formula for the volume of a pyramid. If we take
as the base, then
must be the height.
, so we have an answer of
.
Solution 2 (Bash: One Right Triangle)
We will place tetrahedron in the
-plane. By the Converse of the Pythagorean Theorem, we know that
is a right triangle. Without the loss of generality, let
and
Applying the Distance Formula to
and
respectively, we get the following system:
Subtracting
from
gives
from which
Subtracting from
gives
from which
Substituting into
produces
or
Finally, we find the volume of tetrahedron using
as the base:
~MRENTHUSIASM
Similar Problem
https://artofproblemsolving.com/wiki/index.php/2015_AMC_10A_Problems/Problem_21
Video Solution (Simple & Quick)
~ Education, the Study of Everything
Video Solution (Using Pythagorean Theorem, 3D Geometry - Tetrahedron)
~ pi_is_3.14
Video Solution by TheBeautyofMath
https://youtu.be/t-EEP2V4nAE?t=813
~IceMatrix
See also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.