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| 13 Theory slides |
| 9 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Although not directly used, it can be useful to recall the properties of reflections, rotations, and translations to help identify the differences and similarities of rigid motions and dilations.
On the image you can see a photograph taken in Tennoji Park in Osaka, Japan.
When the vertical slider is moved on the previous applet, the transformation applied to the figure is called a dilation.
OA′=k⋅OA⇔k=OAOA′
The following applet shows the images of points on a straight line. There are two modes in the applet, Setup
and Mark image
— complete the first to reach the next.
This applet shows the distance of two points (middle blue points) and the corresponding image points (further right green points).
The following is a list of a few essential properties of dilations.
The formal proof of this property is beyond the scope of this course. The applet below, however, gives an experimental verification.
The formal proof of this property is beyond the scope of this course. The applet below, however, gives an experimental verification.
Let ∠B′A′C′ be the dilated image of ∠BAC. According to the first property, AC is parallel to A′C′ and AB is parallel to A′B′
Let M be the intersection point of AB and A′C′, and focus on the parallel legs AC and A′C′. According to the Alternate Interior Angles Theorem, angles ∠MAC and ∠AMA′ are congruent.
Similarly, since the other legs are also parallel, angles ∠B′A′M′ and ∠AMA′ are congruent.
Since ∠AMA′ is congruent to both ∠BAC and ∠B′A′C′, the transitive property of congruence implies that these two angles are congruent.
By the definition of congruence, this completes the proof that dilation preserves angle measures.
On the applet below you can experiment with dilations.
Consider the quadrilateral ABCD and the point O in the interior.
For both parts, the properties of dilation guarantee that the image is a quadrilateral, where the vertices are the images of the vertices of a quadrilateral ABCD. These image vertices are on the rays connecting the center of dilation with vertices A, B, C, and D.
The larger logo is an enlarged image of the smaller one.
Scale factor: 1.6
Look for pairs of corresponding points on the two logos.
The center of dilation, a point, and the image of this point are on a straight line. Therefore, the center of dilation can be determined to be the intersection point of any two lines connecting a point and its image.
A ruler can be used to find the distance of any point and its image from the center of dilation.
Use the origin as a center to dilate the triangle by a scale factor 2.
Use horizontal and vertical movements to copy distances.
The points on the coordinate axes stay on the coordinate axes while their distances, from the origin, double.
When moving the third vertex, notice that moving three units to the right and two units up from the origin gives the position of B. The same movement starting at B will end on the ray connecting the origin with B. This image point is twice as far from the origin as the preimage.
Connecting the images of the vertices gives the dilated triangle.
The methods to construct the image of a point depend on the scale factor.
Use a compass to measure the distance from the center of dilation to the preimage point.
To construct the image when the scale factor is 1/n, the properties of dilation can be used.
Use a straightedge to connect the center of dilation with the preimage point and draw a second ray starting at the center of dilation.
Use a compass to copy the same distance n times on the second ray, starting at the center of dilation. The diagram illustrates the process when n=3.
Mark the first and last point on the second ray. Connect the last point with the preimage point and construct a parallel line through the first point.
The combination of the previous two methods gives the dilation of a point by a scale factor p/q.
Depending on the scale factor, choose the appropriate method to construct the image of the vertices.
Segments are drawn between the image vertices to create the image polygon.
The following applet allows you to investigate the effect of applying two dilations one after the other. Use the sliders to adjust the scale factors and move the centers around.
Notice that the combination of the two dilations can be replaced by one dilation.
Consider the polygon ABCDE.
Dilate ABCDE from the origin by a factor of 2 forming A′B′C′D′E′.
Which of the following diagrams shows the dilated figure.
To enlarge ABCDE by a scale factor of 2, we must double the distance between each vertex and the origin. This can be done if we multiply the coordinates of each point by 2. For Part A, we only have to do this for point B. ccc Point & (x,y) & (2 x, 2 y) [0.5em] B & (5,-2) & (10,-4) The dilated point B' has the coordinates (10,-4).
To determine the correctly dilated figure, we must identify the coordinates of the rest of the vertices. ccc Point & (x,y) & (2 x, 2 y) [0.5em] A & (-3,-2) & (-6,-4) C & (5,4) & (20,8) D & (1,6) & (2,12) E & (-3,4) & (-6,8) Now we can plot the dilated polygon.
Comparing A'B'C'D'E' to the given alternatives, we see that iii is the correct option.
For the dilation of A to A′, find the scale factor k.
From the exercise, we have been given the preimage A and the image A'. The scale factor of dilation can be determined by identifying corresponding points on the image and preimage.
If we divide the distance between C and the point on the image by the distance between C and the corresponding point on the preimage, we can determine the scale factor k. k= Distance to imageDistance to preimage Let's do this with the given lengths.
The scale factor is 2.
Like in Part A, we will identify corresponding points on the preimage and image.
As in Part A, we will divide the distance from the center of dilation and the point on the image, by the corresponding length between the center of dilation and the point on the preimage.
The scale factor is 3.
Let's repeat the procedure from previous parts and identify corresponding points.
Having identified corresponding points, we can identify the scale factor.
The scale factor is 13.
Which scale factor(s) create a dilation of AB that is shorter than AB? Select all that apply.
A dilation is a transformation where a figure is either enlarged or reduced. That depends on the scale factor of the dilation. A scale factor between 0 and 1 produces a reduction, and a scale factor greater than 1 gives an enlargement. Therefore, the scale factors that result in a reduction are the following. $1/3$ $1/2$
In which section, 1-12, would you place the center of dilation when dilating P to P′?
To determine the center of dilation, we should draw lines between corresponding points on the preimage and image. Where two of these lines — or more — intersect, we have the center of dilation. To draw one such line, we need to use a straightedge and a pencil.
Let's draw a second line between two more corresponding points.
As we can see, the center of dilation is in section 1.
Like in Part A, we will draw lines between corresponding points on the image and preimage. Where two of them intersect, we have our center of dilation.
Let's draw one more of these lines between two more corresponding points on the preimage and image.
The center of dilation is in section 4.
Let's repeat the procedure once more for Part C.
The center of dilation is in section 6.