ProvingTrianglesSimilar

Similarity Transformations

Dilations Review

• A dilation stretches or shrinks a figure by a determined scale factor

• If the Scale Factor (SF) is greater than 1, it is a stretch, or enlargement.

• If the Scale Factor (SF) is less than 1, it is a shrink, or reduction.

• If the Scale Factor (SF) is equal to 1, there is no change, and the figures will also be congruent.

• On the coordinate plane, a preimage is dilated about a point, which can be the origin

• The point is called the center of dilation

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Drawing a Dilation from a Point

• 1. Draw a line from the center of dilation to a point on the preimage

• Measure the length and multiply that length by the Scale Factor.

• Draw a line from the center of dilation through the original point with the dilated length.

• Mark that point as A¹ (A prime)

How to Draw a Dilation (continued)

• Repeat the process for all of the points in the preimage, denoting the points of the dilated image with "tick" marks to indicate "prime"

• Connect the points to form the new image.

• The new figure should have the shape, but not the same size

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Multiple Choice

What scale factor was used to produce the image?

• 2

• -2

• ¹/₂

• 1

Multiple Choice

State the coordinate of the image of the given point B (4,9) under a dilation with center at the origin with the given scale factor of 2.

• B' (2 , 4.5)

• B' (-8 ,-18)

• B' (8 ,18)

• B' (9 ,4)

Multiple Choice

Similar figures have the same ______ but different ______. 

• measure; shapes
• sizes; shapes
• value; measures
• shape; sizes

Multiple Choice

When the scale factor of a dilation is less than one, the dilation is a reduction.

• True
• False

Multiple Choice

When a dilation has a scale factor greater than one, the dilation is an enlargement.

• True
• False

Multiple Choice

A dilation produces a congruent figure

• True

• False

Multiple Choice

A dilation is which of the following: 

• only an enlargement
• only a reduction
• Either an enlargement or a reduction
• Neither an enlargement or a reduction

Similarity Transformations

• composition (multiple transformations on the same object) of one or more rigid motions (translation, reflection, rotation) and a dilation

Similarity Transformations

• Result in an image that is similar to the original figure (preimage)

• Symbol for similar is  $\sim$  

Similarity Transformations

• All rigid motions are also similarity transformations

• All dilations are similarity transformations

• All compositions of a dilation and a rigid motion are similarity transformations

• Written as DkRxor y-axis or DkT(transformation rule) or Dkr(number of degrees)

Similarity Transformations

• Produce similar figures

• Corresponding angles of the similar figures are the same (the sum of the angles of a triangle = 180^o for all triangles)

• The corresponding sides of a preimage and the resulting image are proportional. The proportion is the Scale Factor

Multiple Choice

ΔOBC is similar to ΔOGT because: (choose the best answer)

• ΔOBC can be dilated about point O by a scale factor of 1/2, and then rotated 90 degrees about point O.

• ΔOBC can be dilated about point G by a scale factor of 1/2, and then reflected over the x- axis

• They aren't similar

• They map to each other with similarity transformations.

Multiple Choice

Quad OBCD is similar to Quad OHTE because:

• Quad OBCD can be reflected over the x axis and then rotated 90 degrees

• Quad OBCD can be reflected over the x-axis and then dilated about point O by a scale factor of 2.

• Quad OBCD can be reflected over the y-axis and then dilated about point O by a scale factor of 2.

• Quad OBCD can be reflected over the y-axis and then dilated about point O by a scale factor of 1/2.

Multiple Select

Pentagon ABCDE is similar to Pentagon RYMNT. Which of the following is correct.

• ∠C ≅ ∠M

• ∠T ≅ ∠E

• ∠Y ≅ ∠B

• ∠A ≅ ∠M

• ∠C ≅ ∠R

Multiple Choice

Why are isometric transformations a part of the similarity transformations?

• Isometric transformations maintain SIZE, and similarity transformations maintain size too.

• Isometric transformations maintain SHAPE, and similarity transformations maintain shape too.

Multiple Choice

What makes isometric transformations different than similarity transformations?

• Isometric transformations have the same shape AND size, similarity transformations just have the same shape.

• Isometric transformations are cooler

• Isometric transformations have the same shape AND size, similarity transformations just have the same size.

• They are the same, no differences.

Multiple Select

Which of the following are similarity transformations? Choose all that apply.

• Translation

• Dilation

• Rotation

• Stretch

• Reflection

Multiple Choice

The image shows what type of transformation

• Rotation

• Translation

• Reflection across the x axis

• Reflection across the y axis

Multiple Choice

A triangle has vertices with coordinates (2,0), (3, -1) and (-2,-5). If the triangle is dilated by a scale factor of 3 with the origin as the center of dilation, what are the coordinates of the vertices of the image?

• (5,3), (6,2), (1,-2)
• (6,0), (9,-3), (-6,-15)
• (2/3,0), (1,-1/3), (-2/3,-5/3)
• (-1,-3), (0,-4), (-5,-8)

Multiple Choice

Determine how to translate triangle ABC to triangle A'B'C'.

• (x-2, y+5)

• (x+2, y-9)

• (x-2, y+9)

• (x-9, y+2)

Multiple Choice

Which of the statements is true about the graphed triangles?

• They are similar, because one can be obtained by dilating the other about the origin with a scale factor of 2.
• They are similar, because one can be obtained by dilating the other about the origin with a scale factor of 1/2.
• They are not similar, because one can be obtained by dilating the other about the origin with a scale factor of 2.
• They are not similar, because one can be obtained by dilating the other about the origin with a scale factor of 1/2.

Multiple Choice

Suppose a triangle with the vertices A(–2, 4), B(–1, –3), and C(4, –2) were dilated, with the center of dilation at the origin and with a scale factor of 2. What will be the coordinates of C'?

• (2, -1)
• (8, -4)
• (8, 4)
• (2, 1)

Multiple Choice

Which series of transformations were performed in the figure?

• Translation 5 units down, reflection over the y-axis

• Reflection over the x-axis, translation 6 units right

• Rotation 90^o clockwise, reflection over the x-axis

• Rotation 180^o clockwise, reflection over the y-axis