Difference between revisions of "2016 AMC 10B Problems/Problem 20"
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real labelscalefactor = 0.5; /* changes label-to-point distance */ | real labelscalefactor = 0.5; /* changes label-to-point distance */ | ||
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ real xmin = -7., xmax = 9., ymin = -7., ymax = 9.6; /* image dimensions */ | pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ real xmin = -7., xmax = 9., ymin = -7., ymax = 9.6; /* image dimensions */ | ||
− | pen xdxdff = rgb(0.49019607843137253,0.49019607843137253,1.); pen uuuuuu = rgb(0. | + | pen xdxdff = rgb(0.49019607843137253,0.49019607843137253,1.); pen uuuuuu = rgb(0.666666666,0.26666666666666666,0.26666666666666666); pen qqzzff = rgb(0.,0.6,1.); pen ffwwqq = rgb(1.,0.4,0.); pen qqwuqq = rgb(0.,0.39215686274509803,0.); |
pair O = (0.,0.), A = (2.,2.), B = (2.,0.), C = (3.209655293155585,3.5927128026548525), D = (2.,4.), F = (-3.999634206191805,-5.999512274922407), G = (-3.999634206191812,-5.9995122749224175); | pair O = (0.,0.), A = (2.,2.), B = (2.,0.), C = (3.209655293155585,3.5927128026548525), D = (2.,4.), F = (-3.999634206191805,-5.999512274922407), G = (-3.999634206191812,-5.9995122749224175); | ||
/* by adihaya */ | /* by adihaya */ | ||
− | draw((2. | + | draw((2.482656878,0.)--(2.482568783,0.48206568779)--(2.,0.48272202065687797)--B--cycle, qqwuqq); |
− | draw((5.482722020656878,0.)--( | + | draw((5.482722020656878,0.)--(7.4827220878,0.48272202065687797)--(5.,0.48272687797)--(5.,0.)--cycle, qqwuqq); |
Label laxis; laxis.p = fontsize(10); | Label laxis; laxis.p = fontsize(10); | ||
xaxis(xmin, xmax, Ticks(laxis, Step = 2., Size = 2, NoZero),EndArrow(6), above = true); | xaxis(xmin, xmax, Ticks(laxis, Step = 2., Size = 2, NoZero),EndArrow(6), above = true); |
Revision as of 17:15, 4 February 2018
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.