Circling Around

Suppose that the circles in the figure below are tangent to one another. The radius of the circle centered at A is one unit, and the radius of the circle centered at B is two units. What are the radii of the congruent circles centered at O and P?

Source: NCTM Problem to Ponder May 2011

SOLUTION

Since the two circles O and P are congruent, the centers of circles A,B,C are collinear. The radius of circle C equals 3. Let r be the radius of circle P\left (x,y\right ).

Distance AP=\sqrt{\left (x-0\right )^2+\left (y-5\right )^2}
1+r=\sqrt{\left (x-0\right )^2+\left (y-5\right )^2}
\left (1+r\right )^2=x^2+\left (y-5\right )^2\qquad\qquad\qquad\text{(1)}

Distance BP=\sqrt{\left (x-0\right )^2+\left (y-2\right )^2}
2+r=\sqrt{\left (x-0\right )^2+\left (y-2\right )^2}
\left (2+r\right )^2=x^2+\left (y-2\right )^2\qquad\qquad\qquad\text{(2)}

Distance CP=\sqrt{\left (x-0\right )^2+\left (y-3\right )^2}
3-r=\sqrt{\left (x-0\right )^2+\left (y-3\right )^2}
\left (3-r\right )^2=x^2+\left (y-3\right )^2\qquad\qquad\qquad\text{(3)}

Multiply Eq. (1) by -1 and add to Eq. (2)
-\left (1+r\right )^2=-x^2-\left (y-5\right )^2
\left (2+r\right )^2=x^2+\left (y-2\right )^2
—————————————
\left (2+r\right )^2-\left (1+r\right )^2=\left (y-2\right )^2-\left (y-5\right )^2
\left (2+r+1+r\right )\left (2+r-1-r\right )=\left (y-2+y-5\right )\left (y-2-y+5\right )
\left (3+2r\right )\left (1\right )=\left (2y-7\right )\left (3\right )
3+2r=6y-21\qquad\qquad\qquad\qquad\qquad\text{(4)}

Multiply Eq. (2) by -1 and add to Eq. (3)
-\left (2+r\right )^2=-x^2-\left (y-2\right )^2
\left (3-r\right )^2=x^2+\left (y-3\right )^2
—————————————–
\left (3-r\right )^2-\left (2+r\right )^2=\left (y-3\right )^2-\left (y-2\right )^2
\left (3-r+2+r\right )\left (3-r-2-r\right )=\left (y-3+y-2\right )\left (y-3-y+2\right )
\left (5\right )\left (1-2r\right )=\left (2y-5\right )\left (-1\right )
5-10r=-2y+5\qquad\qquad\qquad\qquad\text{(5)}

Multiply Eq. (5) by 3 and add to Eq. (4)
3+2r=6y-21
15-30r=-6y+15
——————————
18-28r=-6
-28r=-6-18
-28r=-24
r=\frac{-24}{-28}
r=\frac{6}{7}

Answer: \frac{6}{7}

Advertisements

About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s