2015 AMC 10A Problems/Problem 21
Problem 21
Tetrahedron has , , , , , and . What is the volume of the tetrahedron?
Solution
Let the midpoint of be . We have , and so by the Pythagorean Theorem and . Because the altitude from of tetrahedron passes touches plane on , it is also an altitude of triangle . The area of triangle is, by Heron's Formula, given by
Substituting and performing huge (but manageable) computations yield , so . Thus, if is the length of the altitude from of the tetrahedron, . Our answer is thus and so our answer is .