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July 4, 2020

Tetrahedron Problem | AIME I, 1992 | Question 6

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Tetrahedron.

Tetrahedron Problem - AIME I, 1992


Faces ABC and BCD of tetrahedron ABCD meet at an angle of 30,The area of face ABC=120, the area of face BCD is 80, BC=10. Find volume of tetrahedron.

  • is 107
  • is 320
  • is 840
  • cannot be determined from the given information

Key Concepts


Area

Volume

Tetrahedron

Check the Answer


Answer: is 320.

AIME I, 1992, Question 6

Coordinate Geometry by Loney

Try with Hints


First hint

Area BCD=80=\(\frac{1}{2} \times {10} \times {16}\),

where the perpendicular from D to BC has length 16.

Tetrahedron Problem

Second Hint

The perpendicular from D to ABC is 16sin30=8

[ since sin30=\(\frac{perpendicular}{hypotenuse}\) then height = perpendicular=hypotenuse \(\times\) sin30 ]

Final Step

or, Volume=\(\frac{1}{3} \times 8 \times 120\)=320.

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