Difference between revisions of "2021 AMC 10A Problems/Problem 13"
MRENTHUSIASM (talk | contribs) m (→Solution 2 (Bash: Coordinate Geometry)) |
MRENTHUSIASM (talk | contribs) m (→Solution 2 (Bash: Coordinate Geometry)) |
||
Line 8: | Line 8: | ||
==Solution 2 (Bash: Coordinate Geometry)== | ==Solution 2 (Bash: Coordinate Geometry)== | ||
− | We will place tetrahedron <math>ABCD</math> in the <math>xyz</math>-plane. By the Converse of the Pythagorean Theorem, we know that <math>\triangle ACD</math> is a right triangle. Without the loss of generality, let <math>A=(0,0,0), C=(3,0,0), D=(0,4,0),</math> and <math>B=(x,y,z).</math> Applying the Distance Formula to <math>\overline{BA},\overline{BC},</math> and <math>\overline{BD},</math> respectively, we | + | We will place tetrahedron <math>ABCD</math> in the <math>xyz</math>-plane. By the Converse of the Pythagorean Theorem, we know that <math>\triangle ACD</math> is a right triangle. Without the loss of generality, let <math>A=(0,0,0), C=(3,0,0), D=(0,4,0),</math> and <math>B=(x,y,z).</math> Applying the Distance Formula to <math>\overline{BA},\overline{BC},</math> and <math>\overline{BD},</math> respectively, we get the following system: |
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
x^2+y^2+z^2&=2^2, &(1) \\ | x^2+y^2+z^2&=2^2, &(1) \\ |
Revision as of 13:56, 24 April 2021
Contents
Problem
What is the volume of tetrahedron with edge lengths
,
,
,
,
, and
?
Solution 1 (Observations)
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: Coordinate Geometry)
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.