Similarity Transformations

Key Concepts

• How to draw similar triangle
• Dilation
• Draw dilated image
• Identify the dilation undergone for a particular image

Understanding Dilation 

Dilation is a transformation that stretches or shrinks a figure to create a similar figure. It is a similarity transformation.  

Center of Dilation: A fixed point with respect to which a figure is enlarged or reduced is the center of dilation. 

Scale Factor of Dilation: The scale factor of a dilation is the ratio of a side length of the image to the corresponding side length of the original figure.  

If the scale factor is greater than 1 then the dilated image has enlargement and if the scale factor is less than one dilated image had a reduction. 

If the scale factor is greater than 1 the dilated image stretches and if the dilated image is less than 1 the dilated images Shrinks. 

In the transformation dilation, we get similar images or the images are similar 

Coordinate notation of dilation with respect to the origin is (x, y) ®  (kx, ky)    

Similar triangle construction 

Given a triangle ABC in a coordinate plane. To construct a similar triangle to ABC:  

1. Draw line segments from the origin  passing through the vertices of triangles. 
2. Mark points D, E, F from A,B,C from distance equal to OA,OB,OC respectively. 

3. oin the points D, E, F to get similar triangle DEF. 

Draw a dilation with scale factor 2 for a polygon with vertices K(-1, 0), C(1, 2), U(0, -2) 

Write the rule of dilation for the following figure 

The vertices of the given figure are K(4,0), Z(2,1), M(4,4) and the vertices of the dilated figure are K’(2,0) ,Z’(1,0.5), M’(2,2) 

Rule of dilation is (x, y) ® (1 / 2x,1 / 2y)

Exercise

• Draw a dilation of scale factor 2 with given coordinates J(-1,-1),U(1 ,3), W(0,-2)
• Find the coordinates of dilated image of the points X(4,4), Y(2,-1), Z(4,-1)with scale factor ¼.
• Write the rule for the dilation from U(-2,-1), K(0,2), F(2,-2) to U’(-3,-1.5), K’(0,3), F’(3,-3)
• Find the scale factor of dilation from figure A to Figure B

• Draw a dilation of polygon with vertices A(-2, 1), B(-4, 1), C(-2, 4)  and scale factor k = 2
• If A and B are similar triangles then find the value of m and n

Concept Map

What we have learned

What is transformation?

• Transformation is a change.

What is dilation?

• Dilation is a transformation that stretches or shrinks a figure to create a similar figure. It
• is a similarity transformation.

What is a scale factor?

• The scale factor of a dilation is the ratio of a side length of the image to the corresponding side length of the original figure.

What is the center of dilation?

• A fixed point with respect to which a figure is enlarged or reduced is the center of dilation.