1991 AIME Problems/Problem 14
Problem
A hexagon is inscribed in a circle. Five of the sides have length and the sixth, denoted by
, has length
. Find the sum of the lengths of the three diagonals that can be drawn from
.
Solution
Let ,
, and
.
Ptolemy's Theorem on
gives
, and Ptolemy on
x\cdot z+81^2=y^2
y^2-81y-112\cdot 81=0
y=144
ADEF
81y+81^2=z^2
z=135
x=105
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 |