Difference between revisions of "2017 UNCO Math Contest II Problems/Problem 1"
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<asy> | <asy> | ||
pair A=24*dir(40),B=15*dir(220); | pair A=24*dir(40),B=15*dir(220); | ||
+ | pair C1=(0,0),C2=(52,0); | ||
− | |||
draw(circle(C1,24),black); | draw(circle(C1,24),black); | ||
draw(circle(C2,15),black); | draw(circle(C2,15),black); | ||
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== Solution == | == Solution == | ||
− | <math>12</math> | + | ===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
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
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 |