Difference between revisions of "2021 AMC 10A Problems/Problem 13"
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− | Given the observations from Solution 1, where <math>\triangle ABD, \triangle ABC,</math> and <math>\triangle ABD</math> are right triangles, the base is <math>\triangle ABD</math>. We can apply the information about a trirectangular tetrahedron (all of the face angles are right angles), which states that the volume is <math>V=1/6 ABC</math>, where & | + | Given the observations from Solution 1, where <math>\triangle ABD, \triangle ABC,</math> and <math>\triangle ABD</math> are right triangles, the base is <math>\triangle ABD</math>. We can apply the information about a trirectangular tetrahedron (all of the face angles are right angles), which states that the volume is <math>V=1/6 ABC</math>, where &ABC& are the side lengths. |
<math>AB = 2</math>, <math>AD = 4</math>, <math>BD = 3</math>: The volume of <math>\triangle ABD = 4</math>. | <math>AB = 2</math>, <math>AD = 4</math>, <math>BD = 3</math>: The volume of <math>\triangle ABD = 4</math>. |
Revision as of 21:51, 9 August 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
We apply the Distance Formula to and
respectively:
Subtracting
from
gives
from which
Subtracting from
gives
from which
Substituting into
produces
or
Let the brackets denote areas. 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
Trirectangular Tetrahedron Solution
https://mathworld.wolfram.com/TrirectangularTetrahedron.html
Given the observations from Solution 1, where and
are right triangles, the base is
. We can apply the information about a trirectangular tetrahedron (all of the face angles are right angles), which states that the volume is
, where &ABC& are the side lengths.
,
,
: The volume of
.
The answer is .
-AMC60
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.