Unit 2, Lesson 6
Connecting Similarity and Transformations
Integrated Math 2
6.1
Warm-up
How do you know that is not a dilation of ?
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How do you know that is not a dilation of ?
Are the triangles similar?
Write a sequence of transformations (dilation, translation, rotation, reflection) to take one triangle to the other.
One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure so that it fits exactly over the second. For example, triangle is similar to triangle ().
What is a rotation and a dilation that will take onto ?
The triangles are similar because a rotation of using center will take segment onto segment , because rotations take lines through the center of the rotation to themselves. It will also take segment onto segment for the same reason. Then will be on a ray from through , and will be on a ray from through . Because , a dilation by a scale factor of 2 and center will take to fit exactly onto . Therefore, there is a sequence of rigid motions and dilations that takes onto , and .
Because both rigid motions and dilations leave corresponding angles congruent, all pairs of corresponding angles in similar figures are congruent. In this example, angle is congruent to angle , angle is congruent to angle , and angle is congruent to angle .
Because rigid motions keep lengths congruent and dilations scale them at the same proportion, all pairs of corresponding sides in similar figures are in the same proportion. In this example, .
One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second.
Triangle is similar to triangle because a rotation with center followed by a dilation with center takes to .