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Difference between revisions of "1991 AIME Problems/Problem 14"

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Let <math>x=AC</math>, <math>y=AD</math>, and <math>z=AE</math>.
 
Let <math>x=AC</math>, <math>y=AD</math>, and <math>z=AE</math>.
−
[[Ptolemy's Theorem]] on <math>ABCD</math> gives <math>81y+31\cdot 81=xz</math>, and Ptolemy on <math>ACDE4 gives </math>x\cdot z+81^2=y^2<math>.
+
[[Ptolemy's Theorem]] on <math>ABCD</math> gives <math>81y+31\cdot 81=xz</math>, and Ptolemy on <math>ACDE</math> gives <math>x\cdot z+81^2=y^2</math>.
−
Subtracting these equations give </math>y^2-81y-112\cdot 81=0<math>, and from this </math>y=144<math>. Ptolemy on </math>ADEF<math> gives </math>81y+81^2=z^2<math>, and from this </math>z=135<math>. Finally, plugging back into the first equation gives </math>x=105<math>, so </math>x+y+z=105+144+135=384$.
+
Subtracting these equations give <math>y^2-81y-112\cdot 81=0</math>, and from this <math>y=144</math>. Ptolemy on <math>ADEF</math> gives <math>81y+81^2=z^2</math>, and from this <math>z=135</math>. Finally, plugging back into the first equation gives <math>x=105</math>, so <math>x+y+z=105+144+135=384</math>.
  
 
== See also ==
 
== See also ==

Revision as of 13:05, 21 October 2007

Problem

A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A_{}^{}$.

File:1991 AIME-14.png

Solution

File:1991 AIME-14a.png

Let $x=AC$, $y=AD$, and $z=AE$. Ptolemy's Theorem on $ABCD$ gives $81y+31\cdot 81=xz$, and Ptolemy on $ACDE$ gives $x\cdot z+81^2=y^2$. Subtracting these equations give $y^2-81y-112\cdot 81=0$, and from this $y=144$. Ptolemy on $ADEF$ gives $81y+81^2=z^2$, and from this $z=135$. Finally, plugging back into the first equation gives $x=105$, so $x+y+z=105+144+135=384$.

See also

1991 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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