# Difference between revisions of "2005 AMC 12B Problems/Problem 14"

(added solution) |
|||

Line 1: | Line 1: | ||

== Problem == | == Problem == | ||

− | |||

A circle having center <math>(0,k)</math>, with <math>k>6</math>, is tangent to the lines <math>y=x</math>, <math>y=-x</math> and <math>y=6</math>. What is the radius of this circle? | A circle having center <math>(0,k)</math>, with <math>k>6</math>, is tangent to the lines <math>y=x</math>, <math>y=-x</math> and <math>y=6</math>. What is the radius of this circle? | ||

## Revision as of 23:05, 9 June 2010

## Problem

A circle having center , with , is tangent to the lines , and . What is the radius of this circle?

## Solution

Let be the radius of the circle. Draw the two radii that meet the points of tangency to the lines . We can see that a square is formed by the origin, two tangency points, and the center of the circle. The side lengths of this square are and the diagonal is . The diagonal of a square is times the side length. Therefore, .