Difference between revisions of "2016 AMC 10B Problems/Problem 20"
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==Solution 3: Logic and Geometry== | ==Solution 3: Logic and Geometry== | ||
− | Using the ratios of the circles <math>\frac{3}{2}</math>, we find that the scale factor is <math>1.5</math>. If the origin had not moved, this indicates that the center of the | + | Using the ratios of radii of the circles, <math>\frac{3}{2}</math>, we find that the scale factor is <math>1.5</math>. If the origin had not moved, this indicates that the center of the |
circle would be <math>(3,3)</math>, simply because of <math>(2 \cdot 1.5, 2 \cdot 1.5)</math>. Since the center has moved from <math>(3,3)</math> to <math>(5,6)</math>, we apply the distance formula and get: <math>\sqrt{(6-3)^2 + (5-3)^2} = \sqrt{13}</math>. | circle would be <math>(3,3)</math>, simply because of <math>(2 \cdot 1.5, 2 \cdot 1.5)</math>. Since the center has moved from <math>(3,3)</math> to <math>(5,6)</math>, we apply the distance formula and get: <math>\sqrt{(6-3)^2 + (5-3)^2} = \sqrt{13}</math>. | ||
Revision as of 09:56, 13 October 2017
Contents
Problem
A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius centered at
to the circle of radius
centered at
. What distance does the origin
, move under this transformation?
Solution 1: Algebraic
The center of dilation must lie on the line , which can be expressed
. Also, the ratio of dilation must be equal to
, which is the ratio of the radii of the circles. Thus, we are looking for a point
such that
(for the
-coordinates), and
. Solving these, we get
and
. This means that any point
on the plane will dilate to the point
, which means that the point
dilates to
. Thus, the origin moves
units.
Solution 2: Geometric
Using analytic geometry, we find that the center of dilation is at
and the coefficient/factor is
. Then, we see that the origin is
from the center, and will be
from it afterwards.
Thus, it will move .
Solution 3: Logic and Geometry
Using the ratios of radii of the circles, , we find that the scale factor is
. If the origin had not moved, this indicates that the center of the
circle would be
, simply because of
. Since the center has moved from
to
, we apply the distance formula and get:
.
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.