Solve Problems with Congruence and Similarities

Transformations, Congruence & Similarity

Multiple Choice

Are the two angles below congruent?

• yes

• no

Multiple Choice

Are these triangles similar?

• Yes

• No

​Vocabulary

• Transformation - an operation that maps an original figure onto a new figure ​

• Original Figure - is called the PRE-IMAGE

• New Figure - is called the IMAGE​

• Rigid Transformation - a transformation that creates an image that is congruent to the pre-image. (translation, reflection, and rotation ONLY)​

THESE ARE THE ONLY 3 TRANSFORMATIONS THAT CREATE CONGRUENT IMAGES! RIGID TRANSFORMATIONS

​Types of Transformations

​Translation - a slide that moves all points in the same direction and same distance

​Reflection - a FLIP over a certain line, all points from the pre-image to image are the same distance apart.

​Rotation - a TURN around a fixed point, all points from the pre-image to image are the same distance apart.

Multiple Choice

Fill in the blank: Translations create __________ figures.

• Similar

• Congruent

Multiple Choice

Fill in the blank: Rotations create __________ figures.

• Similar

• Congruent

Multiple Choice

Fill in the blank: Reflections create __________ figures.

• Similar

• Congruent

Multiple Choice

Name the transformation.

• Translation

• Reflection

• Rotation

Multiple Choice

What is this picture an example of? 

• An angle

• A cat vertex 

• Rotation 

Multiple Choice

What transformation is happening to the red triangle to get to the blue triangle?

• Translation
• Rotation
• Reflection

​Intro to Dilations

​Key Concepts:

• A DILATION stretches of shrinks a figure.

  ​

• The CENTER OF DILATION never changes position. It is the starting point.

  ​

• ​The SCALE FACTOR tells you how much the dilation stretches or shrinks.

A DILATION is a transformation that produces an image that is the same shape as the pre-image, but is a different size. Dilations create similar figures. 

• Scale factor is greater than 1​

• Image gets BIGGER​

Enlargement

• Scale factor is less than 1​

• ​Image gets SMALLER

​​Reduction

Multiple Choice

First, identify the old and new image. Now, did the triangle get bigger or smaller?

• The triangle got smaller

• The triangle got bigger

Multiple Choice

First, identify the old and new image. Now, did the triangle get bigger or smaller?

• The triangle got smaller

• The triangle got bigger

Multiple Choice

Is this an enlargement or a reduction?

• Enlargement

• Reduction

Multiple Choice

A dilation always produces a similar figure. Similar figures have the same ______ but different ______. 

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

Multiple Choice

A scale factor of 1/2 reduces a figure.

• True
• False

Multiple Choice

A dilation that increases in size will always have a scale factor that is...

• negative

• exactly 1

• greater than 1

• less than 1

Multiple Choice

Triangle ABC is dilated by a scale factor of $$8$$ to form Triangle A’B’C’ with the origin as a center of dilation.

Which rule represents this dilation? 

• $$(8x,8y)\ \rightarrow\ (x,y)\$$  

• $$(x,y)\ \ \rightarrow\ \ (x+8,\ y-8​)$$  
  

• $$(x,y)\ \rightarrow\ (8x,\ 8y)$$  

• 
  

  $$(x,y)\ \ \rightarrow\ \ (\frac{x}{8},\ \frac{y}{8})$$  

Multiple Select

Which of the following algebraic rules represents an enlargement?

***There's more than 1 correct answer!

• $$(x,y)\ \rightarrow\ (1.6x,1.6y)$$  
  

• $$(x,y)\ \rightarrow\ (0.75x,0.75y)$$  
  

• $$(x,y)\ \rightarrow\ (\frac{18}{5}\ x,\ \frac{18}{5}\ y)$$  
  

• $$(x,y)\ \rightarrow\ (-x,y)$$  
  

Multiple Select

Which of the following algebraic rules represents a reduction?

***There's more than 1 correct answer!

• $$(x,y)\ \rightarrow\ (3x,3y)$$  
  

• $$(x,y)\ \rightarrow\ (\frac{3}{4}x,\ \frac{3}{4}​y\ )$$  
  

• $$(x,y)\ \rightarrow\ (0.99x,0.99y)$$  
  

• $$(x,y)\ \rightarrow\ (5x,\ 5y)$$  
  

Multiple Choice

Which rule best represents the dilation that was applied to Pentagon EFGHI to create Pentagon E'F'G'H'I’?

• $$(x,y)\rightarrow\ (x+2,y-4)$$

• $$(x,y)\ \rightarrow\ (0.33x,0.33y)$$

• $$(x,y)\ \rightarrow\ (3x,3y)$$

• $$(x,y)\ \rightarrow\ (x-2,y+4)$$

Multiple Choice

Trapezoid ABCD was dilated with the origin as the center to create Trapezoid A'B'C'D’.

Which rule best represents the dilation that was applied to Trapezoid ABCD to create Trapezoid A'B'C'D' ?

• $$(x,y)\ \ \rightarrow\ \ (\frac{1}{2}x,\frac{1}{2}​y)$$

• $$(x,y)\ \ \rightarrow\ \ (2x,\ 2y)$$
  

• $$(x,y)\ \ \rightarrow\ \ (4x,\ 2​y)$$
  

• $$(x,y)\ \ \rightarrow\ \ (-x,\ -y)$$
  

Multiple Choice

Quadrilateral PNDM was dilated with the origin as the center to create quadrilateral P'N'D'M’.

Which rule best represents the dilation that was applied to quadrilateral PNDM to create quadrilateral P'N'D'M’?

• $$(x,y)\ \rightarrow\ (x-1,\ y+1)$$  

• $$(x,y)\ \rightarrow\ \left(\frac{1}{9}x,\ \frac{1}{9}y)\right)$$  

• $$(x,y)\ \rightarrow\ (0.77x,\ 0.77y)$$

• $$(x,y)\ \rightarrow\ \left(\frac{6}{3}x,\ \frac{6}{3}y)\right)$$  

Multiple Choice

Which transformation results in a figure that is similar to the original figure but has a greater area?

• A dilation of triangle QRS by a scale factor of 0.25

• A dilation of triangle QRS by a scale factor of 0.5

• A dilation of triangle QRS by a scale factor of 1

• A dilation of triangle QRS by a scale factor of 2

Multiple Choice

In the coordinate plane, segment PQ is the result of a dilation of segment XY by a scale factor of 1/2. Which point is the center of dilation?

• (-4, 0)

• (0, -4)

• (0, 4)

• (4, 0)

Multiple Choice

Look at quadrilateral QRST. What is the image of point R after a counterclockwise rotation of 270 degrees about the origin?

• (6, -3)

• (-3, 6)

• (-6, 3)

• (3, -6)

Multiple Choice

Which transformation of HIJ does NOT result in a congruent triangle?

• a reflection across the x axis, followed by a rotation of 180° about the origin

• a rotation by 180° about the origin, followed by a translation of 2 units left & 3 down

• a translation of 1 unit right and 2 units up, followed by a dilation by a factor of 3

• a dilation by a factor of 2, followed by a dilation by a factor of ½

Multiple Choice

Parallelogram FGHJ was translated 3 units down to form parallelogram F’G’H’J’. Parallelogram F’G’H’J’ was then rotated 90° counterclockwise about point G’ to obtain F’’G’’H’’J’’. Which statement is true about parallelogram FGHJ and parallelogram F”G”H”J”?

• The figures are both similar and congruent.

• The figures are neither similar nor congruent.

• The figures are similar but not congruent.

• The figures are congruent but not similar.

Multiple Choice

The __________ is the ratio of a length of the new figure to the corresponding length on the original figure.

• reduction

• enlargement

• scale factor

• center of dilation