1983 AIME Problems/Problem 4: Difference between revisions
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== Problem == | == Problem == | ||
A machine shop cutting tool is in the shape of a notched circle, as shown. The radius of the circle is 50 cm, the length of <math>AB</math> is 6 cm, and that of <math>BC</math> is 2 cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math>B</math> to the center of the circle. | A machine shop cutting tool is in the shape of a notched circle, as shown. The radius of the circle is 50 cm, the length of <math>AB</math> is 6 cm, and that of <math>BC</math> is 2 cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math>B</math> to the center of the circle. | ||
[[Image: | [[Image:AIME_83_-4.JPG]] | ||
== Solution == | == Solution == | ||
Revision as of 01:39, 21 January 2007
Problem
A machine shop cutting tool is in the shape of a notched circle, as shown. The radius of the circle is 50 cm, the length of 
is 6 cm, and that of 
is 2 cm. The angle 
is a right angle. Find the square of the distance (in centimeters) from 
to the center of the circle.
Solution
Because we are given a right angle, we look for ways to apply the Pythagorean Theorem. Extend a perpendicular from 
to and label it 
. Additionally, extend a perpendicular from to the line , and label it 
. Let 
and 
. We're trying to find 
.
Applying the Pythagorean Theorem, 
, and 
.
Thus, 
, and 
. We solve this system to get 
and 
, resulting in an answer of 
.
