group 1- dilation

7/30/2019 Group 1- Dilation

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Transformation Geometry - Dr.Rochmad, M.Si.


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Group I

1. Shofiayuningtyas L. Y (4101410002)

2. Titis Satitis ( 4101410013)

3. Chandra Septian B. ( 4101410030)

4. Endang Nurliastuti ( 4101410031)

Lecturer : Dr.Rochmad, M.Si.


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Dilation EnlargementEnlargement with

center the originReduction

A dilation is a transformation (notation ) thatproduces 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 inthe plane about which all points are expanded

or contracted. It is the only invariant point

under a dilation.

A dilation of scalar factor kwhose center of dilation is theorigin may be written: Dk(x, y) = (kx, ky).


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Dilation of point P(x, y) with dilation [0,k] so that

we get P(x,y) can be formed becomes as follows

= 00

When we have dilation of point P(x, y) with

dilation [(a,b),k] so that we get P(x, y) can be

formed becomes as follows

= 00

+

where 00

as dilation matrix


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

shrinks the original figure.

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 in all cases except a scale factor 1 and -1)

Properties preserved (invariant) under a

dilation:


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Scalar multiplication by a is also geometrically interesting, because itrepresents magnification by the factor k. It magnifies, or dilates, the whole

plane by the factor a, transforming each figure into a similar copy of itself.Figure (a) below shows an example of this with k = 2.5. Figure (b) shows anexample dilation with k = 1/2, which the circle with center M dilated to thecircle with center O.

Figure (a) Figure (b)


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There are two types of dilation, namely enlargement and reduction. If

k > 1, the image figure is an enlargement of the object. If 0< k < 1,

the image figure is a reduction of the object.




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The diagram below shows the enlargement of triangle PQR with center

point C and scale factor k = 3.

Points P,Q and R are located such

that CP = 3 x CP, CQ = 3 x CQ, and

CR = 3 x CR. The image PQR hassides which are 3 times longer than

those of the object PQR.




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Suppose P(x, y) moves to P(x, y) such that P lies on the line OP, andOP = kOP. We call this an dilation with centerO(0,0) and scale factor k.

Under an dilation with centerO(0, 0) and scale factor k, (x, y) (kx, ky).




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A long side is a reduction of triangle KLM with center C and scale factor k =

To obtain the image, the distance from C to each point on the object is halved.




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In the example below, ABC isreduction by a factor of 0,5 through the

origin to obtain ABC.

Instead ABC is similar to ABC. That means that the lengths of AB, AC and BC

have all changed by the same factor.

In the example beside, the image is notcongruent to the original triangle (so, not anisometric).




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In this case, the factor of the reduction is 0,5 and therefore we can say that

each side of ABCis half the length of the corresponding side of ABC. For

example, AB = 2 and the corresponding side AB = 1. Also, because the

shapes are similar, we see that the interior angles remain unchanged. For

example, angle B = 90 and angle B = 90. We say the shape has been

preserved but not the size.




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2. PROBLEM: Draw the dilation image of triangleABC with the centerof 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!


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3. PROBLEM: Draw the dilation image of pentagonABCDEwith the centerof dilation at the origin and a scale factor of 1/3. OBSERVE: Notice how

EVERY coordinate of the original pentagon has been multiplied by thescale factor (1/3).

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


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4. PROBLEM: Draw the dilation image of rectangle EFGHwith the centerof dilation at point Eand a scale factor of 1/2.

OBSERVE: Point Eand its image are the same. It is important to observethe 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.


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5. If the points P (2,-1) is dilated with (O,-2), then the image is.....

Solution

=2 00 2

21

=42


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1. Triangle PQR with P(2,3), Q(2,5) and R(4,7). Sketch the triangle PQR

under:

a. dilation with scale factor k = 3, the center is P.

b. dilation with scale factor k = -4, the center is Q.

c. dilation with scale factor k = , he center is R.

F. Competence Test


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2. Look at the figure below, IHJK a square and L, M, N and O middle of edges.


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3. Given point A ( 5,4) and P (1,2). The image of point A where the dilation is

stated in [P,3] is...


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Group 1

group 1- dilation

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