MathDB

18

Part of 2008 AMC 12/AHSME

Problems(2)

Origin-Triangle Tetrahedron Volume

Source: AMC 12 2008A Problem 18

2/17/2008
Triangle ABC ABC, with sides of length 5 5, 6 6, and 7 7, has one vertex on the positive x x-axis, one on the positive y y-axis, and one on the positive z z-axis. Let O O be the origin. What is the volume of tetrahedron OABC OABC? <spanclass=′latex−bold′>(A)</span> 85<spanclass=′latex−bold′>(B)</span> 90<spanclass=′latex−bold′>(C)</span> 95<spanclass=′latex−bold′>(D)</span> 10<spanclass=′latex−bold′>(E)</span> 105 <span class='latex-bold'>(A)</span>\ \sqrt{85} \qquad <span class='latex-bold'>(B)</span>\ \sqrt{90} \qquad <span class='latex-bold'>(C)</span>\ \sqrt{95} \qquad <span class='latex-bold'>(D)</span>\ 10 \qquad <span class='latex-bold'>(E)</span>\ \sqrt{105}
geometry3D geometrytetrahedronpyramidcalculusanalytic geometryarea of a triangle
Pyramid

Source: AMC 12 2008B Problem 18

2/29/2008
A pyramid has a square base ABCD ABCD and vertex E E. The area of square ABCD ABCD is 196 196, and the areas of △ABE \triangle{ABE} and △CDE \triangle{CDE} are 105 105 and 91 91, respectively. What is the volume of the pyramid? <spanclass=′latex−bold′>(A)</span> 392<spanclass=′latex−bold′>(B)</span> 1966<spanclass=′latex−bold′>(C)</span> 3922<spanclass=′latex−bold′>(D)</span> 3923<spanclass=′latex−bold′>(E)</span> 784 <span class='latex-bold'>(A)</span>\ 392 \qquad <span class='latex-bold'>(B)</span>\ 196\sqrt{6} \qquad <span class='latex-bold'>(C)</span>\ 392\sqrt2 \qquad <span class='latex-bold'>(D)</span>\ 392\sqrt3 \qquad <span class='latex-bold'>(E)</span>\ 784
geometry3D geometrypyramidarea of a triangleHeron's formulaAMC