Difference between revisions of "2021 AMC 10A Problems/Problem 13"
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<math>\textbf{(A)} ~3 \qquad\textbf{(B)} ~2\sqrt{3} \qquad\textbf{(C)} ~4\qquad\textbf{(D)} ~3\sqrt{3}\qquad\textbf{(E)} ~6</math> | <math>\textbf{(A)} ~3 \qquad\textbf{(B)} ~2\sqrt{3} \qquad\textbf{(C)} ~4\qquad\textbf{(D)} ~3\sqrt{3}\qquad\textbf{(E)} ~6</math> | ||
| − | ==Solution== | + | ==Solution 1 (Observations)== |
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: 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> | ||
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| + | ~MRENTHUSIASM | ||
==Similar Problem== | ==Similar Problem== | ||
Revision as of 10:57, 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 
~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. 
