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Topic: Geometry Question #3
Replies: 39   Last Post: Jun 18, 2009 5:05 PM

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 reuterg@canandaiguaschools.org Posts: 15 Registered: 2/18/05
Re: Geometry Question #3
Posted: Jun 17, 2009 10:34 AM
 att1.html (5.4 K)

Dolores,

With respect, regentsprep.org isn't the only source of mathematically valid information. I could quote various sources (mathworld.com, icoachmath.com, etc.) that would say that a dilation could have a factor of 1 or -1 (even though we recognize these would be trivial dilations).

I do, however, agree that there's a lot of guesswork a student had to do to answer the question, and that the guesswork that would have led to "rotation" (realizing that ABC is congruent to A'B'C') could just as easily led that student to "dilation" (although the savvy student probably picked rotation because it was choice 1).

George Reuter

>>> "Storey, Dolores" <DStorey@newlebanoncsd.org> 6/17/2009 9:32 AM >>>
Here is the information from regentsprep.org

Dilations
Topic Index | Geometry Index | Regents Exam Prep Center

A dilation is a transformation (notation ) that produces an image that is the same shape as the original, but is a different size. A dilation stretches or shrinks the original figure.

The description of a dilation includes the scale factor (or ratio) and the center of the dilation. The center of dilation is a fixed point in the plane about which all points are expanded or contracted. It is the only invariant point under a dilation.

A dilation of scalar factor k whose center of dilation is the origin
may be written: Dk (x, y) = (kx, ky).
If the scale factor, k, is greater than 1, the image is an enlargement (a stretch).
If the scale factor is between 0 and 1, the image is a reduction (a shrink).
(If the scale factor should be less than 0, a dilation has occurred as well as a reflection in the center.)

Properties preserved (invariant) under a dilation:
1. angle measures (remain the same)
2. parallelism (parallel lines remain parallel)
3. colinearity (points stay on the same lines)
4. midpoint (midpoints remain the same in each figure)
5. orientation (lettering order remains the same)
- ---------------------------------------------------------------
6. distance is NOT preserved (NOT an isometry)
(lengths of segments are NOT the same)
Dilations create similar figures.

Definition: A dilation is a transformation of the plane, , such that if O is a fixed point, k is a non-zero real number, and P' is the image of point P, then O, P and P' are collinear and .
Notation:

Examples:1.

P' is the image of P under a
dilation about O of ratio 2.
OP' = 2OP and

2.
is the image of under a dilation about O of ratio .

Most dilations in coordinate geometry use the origin, (0,0), as the center of the dilation.

Example 1:

PROBLEM: Draw the dilation image of triangle ABC with the center of dilation at the origin and a scale factor of 2.
OBSERVE: Notice how EVERY coordinate of the original triangle has been multiplied by the scale factor (x2).

HINT: Dilations involve multiplication!

Example 2:

PROBLEM: Draw the dilation image of pentagon ABCDE with the center of dilation at the origin and a scale factor of 1/3.
OBSERVE: Notice how EVERY coordinate of the original pentagon has been multiplied by the scale factor (1/3).

HINT: Multiplying by 1/3 is the same as dividing by 3!

For this example, the center of the dilation is NOT the origin. The center of dilation is a vertex of the original figure.

Example 3:

PROBLEM: Draw the dilation image of rectangle EFGH with the center of dilation at point E and a scale factor of 1/2.
OBSERVE: Point E and its image are the same. It is important to observe the distance from the center of the dilation, E, to the other points of the figure. Notice EF = 6 and E'F' = 3.

HINT: Be sure to measure distances for this problem.

- --------------------------------------------------------------------------------

Topic Index | Geometry Index | Regents Exam Prep Center
Created by Donna Roberts
Copyright 1998-2009 http://regentsprep.org ( http://regentsprep.org/ )
Oswego City School District Regents Exam Prep Center

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