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

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== Problem ==
== 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 ==
== 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]], <math>O_1P</math> is <math>\frac{24}{39}\cdot 52</math> and <math>O_2P</math> is <math>\frac{15}{39}*52</math>. Their difference is <math>\frac{9}{39}\cdot 52</math>, or <math>\boxed{12}</math>


== See also ==
== See also ==

Latest revision as of 17:16, 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 $\boxed{12}$

See also

2017 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by
First Question 		Followed by
Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNCO Math Contest Problems and Solutions
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