2021 AMC 10A Problems/Problem 13
Contents
- 1 Problem
- 2 Solution 1 (Three Right Triangles)
- 3 Solution 2 (Bash: One Right Triangle)
- 4 Solution 3 (Trirectangular Tetrahedron)
- 5 Solution 4
- 6 Remark
- 7 Video Solution (Simple & Quick)
- 8 Video Solution (Using Pythagorean Theorem, 3D Geometry: Tetrahedron)
- 9 Video Solution by TheBeautyofMath
- 10 See also
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 altitude. The volume of tetrahedron is
~Icewolf10 ~Bakedpotato66 ~MRENTHUSIASM
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
Solution 3 (Trirectangular Tetrahedron)
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 ~AMC60 (Solution)
~MRENTHUSIASM (Revision)
Solution 4
Notice that and are right angles. Let the base of the tetrahedron be , so the height is . Therefore,
~ kante314
Remark
Here is a similar problem from another AMC test: 2015 AMC 10A 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 |
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