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Difference between revisions of "2001 AIME I Problems/Problem 12"

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m (Fixed asy)
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== Solution ==
 
== Solution ==
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<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);
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pair A=(6,0,0), B=(0,4,0), C=(0,0,2), D=(0,0,0);
+
triple I = (3/2,1,1/2);
−
D(MP("A",A)--MP("B",B)--MP("C",C,N)--A--MP("D",D)--B--D--C);
+
draw(C--A--D--C--B--D--I--A--B--I--C);
−
</asy></center>
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label("$I$",I,S);
 +
label("$A$",A,S);
 +
label("$B$",B,E);
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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>\begin{align*}V &= V_{ABCI}+V_{ABDI}+V_{ACDI}+V_{BCDI}\\ &= \frac 13 \cdot r \cdot \left([ABC] + [ABD] + [ACD] + [BCD]\right) = \frac{rS}{3}, \end{align*}</math></center> where <math>S</math> is the [[surface area]] of <math>ABCD</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>\begin{align*}V &= V_{ABCI}+V_{ABDI}+V_{ACDI}+V_{BCDI}\\ &= \frac 13 \cdot r \cdot \left([ABC] + [ABD] + [ACD] + [BCD]\right) = \frac{rS}{3}, \end{align*}</math></center> where <math>S</math> is the [[surface area]] of <math>ABCD</math>.

Revision as of 09:42, 12 March 2014

Problem

A sphere is inscribed in the tetrahedron whose vertices are $A = (6,0,0), B = (0,4,0), C = (0,0,2),$ and $D = (0,0,0).$ The radius of the sphere is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Solution

[asy]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); draw(C--A--D--C--B--D--I--A--B--I--C); label("$I$",I,S); label("$A$",A,S); label("$B$",B,E); label("$C$",C,N); label("$D$",D,W);[/asy]

Connect all four vertices of tetrahedron $ABCD$ to its incenter, $I$. This yields four tetrahedra $ABCI, ABDI, ACDI, BCDI$, all of which have height of $r$ (the radius of the inscribed sphere), and which together form $ABCD$. It follows that

$\begin{align*}V &= V_{ABCI}+V_{ABDI}+V_{ACDI}+V_{BCDI}\\ &= \frac 13 \cdot r \cdot \left([ABC] + [ABD] + [ACD] + [BCD]\right) = \frac{rS}{3}, \end{align*}$ (Error compiling LaTeX. Unknown error_msg)

where $S$ is the surface area of $ABCD$.

Since $\triangle ABD, ACD, BCD$ all lie on the planes containing the axes, their areas are straightforward to calculate; respectively $12,6,4$. To find $[ABC]$, we can use the 3-dimensional distance formula ($d=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}$) to find that $AB = \sqrt{52}, BC=\sqrt{20}, CA=\sqrt{40}$. From here, we can use the Law of Cosines and the sine area formula to compute $[ABC]$, or we can use a manipulated version of Heron's formula: $A = \frac{1}{4}\sqrt{4a^2b^2 - (a^2+b^2-c^2)^2} = 14$.[1]

Thus, $S = 14 + 12 + 6 + 4 = 36$. The volume of $ABCD$ we can compute by letting $AD$ to be the height to face $BCD$, so $V = \frac{1}{3} \cdot 6 \cdot \frac 12 \cdot 4 \cdot 2 = 8$. Therefore, $r = \frac{3V}{S} = \frac{24}{36} = \frac 23$, and $m+n = \boxed{005}$.


^ There are a couple of other ways to compute $[ABC]$, including by vectors. In fact, it is known that in a trirectangular tetrahedron (one in which three edges are mutually perpendicular, as is the case here), the sum of the squares of the areas of the three smaller faces equals the square of the area of the larger face. See the thread below for details.

See also

  • <url>viewtopic.php?p=384205#384205 Discussion on AoPS</url>
2001 AIME I (Problems • Answer Key • Resources)
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Problem 13
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All AIME Problems and Solutions

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