2012 AMC 10B Problems/Problem 12
Revision as of 13:15, 4 July 2013 by Nathan wailes (talk | contribs)
Problem
Point B is due east of point A. Point C is due north of point B. The distance between points A and C is , and
. Point D is 20 meters due north of point C. The distance AD is between which two integers?
Solution
If point B is due east of point A and point C is due north of point B, is a right angle. And if
,
is a 45-45-90 triangle. Thus, the lengths of sides
,
, and
are in the ratio
, and
is
.
is clearly a right triangle with
on the side
.
is 20, so
.
By the Pythagorean Theorem, .
, and
. Thus,
must be between
and
. The answer is
.
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.