Difference between revisions of "2002 AMC 12P Problems/Problem 8"
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== Problem == | == Problem == | ||
| − | + | Let <math>AB</math> be a segment of length <math>26</math>, and let points <math>C</math> and <math>D</math> be located on <math>AB</math> such that <math>AC=1</math> and <math>AD=8</math>. Let <math>E</math> and <math>F</math> be points on one of the semicircles with diameter <math>AB</math> for which <math>EC</math> and <math>FD</math> are perpendicular to <math>AB</math>. Find <math>EF.</math> | |
| − | <math> \ | + | <math> |
| + | \text{(A) }5 | ||
| + | \qquad | ||
| + | \text{(B) }5 \sqrt{2} | ||
| + | \qquad | ||
| + | \text{(C) }7 | ||
| + | \qquad | ||
| + | \text{(D) }7 \sqrt{2} | ||
| + | \qquad | ||
| + | \text{(E) }12 | ||
| + | </math> | ||
| + | |||
| + | [[2002 AMC 12P Problems/Problem 8|Solution]] | ||
== Solution == | == Solution == | ||
Revision as of 23:45, 29 December 2023
Problem
Let 
be a segment of length 
, and let points 
and 
be located on such that 
and 
. Let 
and 
be points on one of the semicircles with diameter for which 
and 
are perpendicular to . Find 
Solution
If 
, then 
. Since 
, 
must be 
to some factor of 6. Thus, there are four (3, 9, 27, 729) possible values of 
.
See also
| 2002 AMC 12P (Problems • Answer Key • Resources) | |
| Preceded by Problem 7 |
Followed by Problem 9 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
