Difference between revisions of "2016 AMC 10B Problems/Problem 20"
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− | The center of dilation must lie on the line <math>A A'</math>, which can be expressed as <math>y = \dfrac{4x}{3} - \dfrac{2}{3}</math>. Note that the center of dilation must have an <math>x</math>-coordinate less than <math>2</math> | + | The center of dilation must lie on the line <math>A A'</math>, which can be expressed as <math>y = \dfrac{4x}{3} - \dfrac{2}{3}</math>. Note that the center of dilation must have an <math>x</math>-coordinate less than <math>2</math>; if the <math>x</math>-coordinate were otherwise, than the circle under the transformation would not have an increased <math>x</math>-coordinate in the coordinate plane. Also, the ratio of dilation must be equal to <math>\dfrac{3}{2}</math>, which is the ratio of the radii of the circles. Thus, we are looking for a point <math>(x,y)</math> such that <math>\dfrac{3}{2} \left( 2 - x \right) = 5 - x</math> (for the <math>x</math>-coordinates), and <math>\dfrac{3}{2} \left( 2 - y \right) = 6 - y</math>. We do not have to include absolute value symbols because we know that the center of dilation has a lower <math>x</math>-coordinate, and hence a lower <math>y</math>-coordinate, from our reasoning above. Solving the two equations, we get <math>x = -4</math> and <math>y = - 6</math>. This means that any point <math>(a,b)</math> on the plane will dilate to the point <math>\left( \dfrac{3}{2} (a + 4) - 4, \dfrac{3}{2} (b + 6) - 6 \right)</math>, which means that the point <math>(0,0)</math> dilates to <math>\left( 6 - 4, 9 - 6 \right) = (2,3)</math>. Thus, the origin moves <math>\sqrt{2^2 + 3^2} = \boxed{\sqrt{13}}</math> units. |
===Solution 2: Geometric=== | ===Solution 2: Geometric=== |
Revision as of 07:08, 28 October 2019
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?
Solutions
Solution 1: Algebraic
The center of dilation must lie on the line , which can be expressed as
. Note that the center of dilation must have an
-coordinate less than
; if the
-coordinate were otherwise, than the circle under the transformation would not have an increased
-coordinate in the coordinate plane. 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
. We do not have to include absolute value symbols because we know that the center of dilation has a lower
-coordinate, and hence a lower
-coordinate, from our reasoning above. Solving the two equations, 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:
.
Solution 4: Simple and Practical
Start with the size transformation. Transforming the circle from radius 2 to radius 3 would mean the origin point now transforms into the point (-1,-1). Now apply the position shift; 3 to the right and 4 up which gets you the pt (2,3). Now simply do Pythagorean theorem with pts (0,0) and (2,3) to find the distance traveled.
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 |
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