Difference between revisions of "2017 UNCO Math Contest II Problems/Problem 1"
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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 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]](http://latex.artofproblemsolving.com/1/0/c/10cd3983047460719a87d83accaf160349109a91.png)
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]](http://latex.artofproblemsolving.com/b/d/a/bdaa193d3b8004af6e044060ac20b5760d8604bb.png)
Solution 1
By similar triangles, 
is 
and 
is 
. Their difference is 
, or 
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 | ||
