Difference between revisions of "2017 UNCO Math Contest II Problems/Problem 1"

Latest revision as of 18:16, 16 January 2023

Problem

$[asy] pair A=24*dir(40),B=15*dir(220); pair C1=(0,0),C2=(52,0); draw(circle(C1,24),black); draw(circle(C2,15),black); draw(C1--C2,dot); draw(A--(B+C2)); [/asy]$

A circle has radius 24, a second circle has radius 15, and the centers of the two circles are 52 units apart. A line tangent to both circles crosses the line connecting the two centers at a point P between the two centers. How much farther is P from the center of the bigger circle than it is from the center of the smaller circle?

Solution

Diagram

$[asy] pair A=24*dir(40), B=15*dir(220); pair C1=(0,0),C2=(52,0); pair [] x=intersectionpoints(C1--C2, A--(B+C2)); pair P=x[0]; draw(circle(C1,24),black); draw(circle(C2,15),black); draw(C1--C2,dot); draw(A--(B+C2), dot); draw(P--P, dot); draw(C1--A); draw(C2--(B+C2)); label("A",A,NE); label("B",B+C2,SW); label("O1",C1,S); label("O2",C2,N); label("P",P,SW); [/asy]$

Solution 1

By similar triangles, $O_1P$ is $\frac{24}{39}\cdot 52$ and $O_2P$ is $\frac{15}{39}*52$ . Their difference is $\frac{9}{39}\cdot 52$ , or $\boxed{12}$

2017 UNCO Math Contest II (Problems • Answer Key • Resources)
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