Difference between revisions of "2021 AMC 10A Problems/Problem 13"
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| − | + | ==Problem== | |
| + | What is the volume of tetrahedron <math>ABCD</math> with edge lengths <math>AB = 2</math>, <math>AC = 3</math>, <math>AD = 4</math>, <math>BC = \sqrt{13}</math>, <math>BD = 2\sqrt{5}</math>, and <math>CD = 5</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== | ||
| + | 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>. | ||
| + | |||
| + | ==Similar Problem== | ||
| + | https://artofproblemsolving.com/wiki/index.php/2015_AMC_10A_Problems/Problem_21 | ||
| + | |||
| + | ==Video Solution (Simple & Quick)== | ||
| + | https://youtu.be/bRrchiDCrKE | ||
| + | |||
| + | ~ Education, the Study of Everything | ||
| + | |||
| + | == Video Solution (Using Pythagorean Theorem, 3D Geometry - Tetrahedron) == | ||
| + | https://youtu.be/i4yUaXVUWKE | ||
| + | |||
| + | ~ pi_is_3.14 | ||
| + | |||
| + | ==Video Solution by TheBeautyofMath== | ||
| + | https://youtu.be/t-EEP2V4nAE?t=813 | ||
| + | |||
| + | ~IceMatrix | ||
| + | |||
| + | ==See also== | ||
| + | {{AMC10 box|year=2021|ab=A|num-b=12|num-a=14}} | ||
| + | {{MAA Notice}} | ||
Revision as of 11:10, 4 April 2021
Contents
Problem
What is the volume of tetrahedron 
with edge lengths 
, 
, 
, 
, 
, and 
?
Solution
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 
.
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. 
