Difference between revisions of "2001 AIME I Problems/Problem 12"
m (→Solution) |
m (Fixed asy) |
||
Line 3: | Line 3: | ||
== Solution == | == Solution == | ||
− | <center><asy> | + | <center><asy>import three; pointpen = black; pathpen = black+linewidth(0.7); currentprojection = perspective(5,-10,4); |
− | import three; pointpen = black; pathpen = black+linewidth(0.7); currentprojection = perspective(5,-10,4); | + | triple A = (6,0,0), B = (0,4,0), C = (0,0,2), D = (0,0,0); |
− | + | triple I = (3/2,1,1/2); | |
− | D( | + | draw(C--A--D--C--B--D--I--A--B--I--C); |
− | </asy></center> | + | label("$I$",I,S); |
+ | label("$A$",A,S); | ||
+ | label("$B$",B,E); | ||
+ | label("$C$",C,N); | ||
+ | label("$D$",D,W);</asy></center> | ||
Connect all four vertices of tetrahedron <math>ABCD</math> to its [[incenter]], <math>I</math>. This yields four tetrahedra <math>ABCI, ABDI, ACDI, BCDI</math>, all of which have height of <math>r</math> (the radius of the inscribed sphere), and which together form <math>ABCD</math>. It follows that <center><math> | Connect all four vertices of tetrahedron <math>ABCD</math> to its [[incenter]], <math>I</math>. This yields four tetrahedra <math>ABCI, ABDI, ACDI, BCDI</math>, all of which have height of <math>r</math> (the radius of the inscribed sphere), and which together form <math>ABCD</math>. It follows that <center><math> |
Revision as of 09:42, 12 March 2014
Problem
A sphere is inscribed in the tetrahedron whose vertices are and The radius of the sphere is where and are relatively prime positive integers. Find
Solution
Connect all four vertices of tetrahedron to its incenter, . This yields four tetrahedra , all of which have height of (the radius of the inscribed sphere), and which together form . It follows that
where is the surface area of .
Since all lie on the planes containing the axes, their areas are straightforward to calculate; respectively . To find , we can use the 3-dimensional distance formula () to find that . From here, we can use the Law of Cosines and the sine area formula to compute , or we can use a manipulated version of Heron's formula: .[1]
Thus, . The volume of we can compute by letting to be the height to face , so . Therefore, , and .
See also
- <url>viewtopic.php?p=384205#384205 Discussion on AoPS</url>
2001 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.