G8 Practices 9-3 Similarity and Transformations

Are the two figures similar? If so, describe a sequence of transformations that maps one figure onto the other figure. If not, explain why they are not similar.

• similar; Dilate rectangle ABCD using a scale factor of 0.5 and center of dilation at the origin, and then reflect it over the y-axis.

• similar; Dilate rectangle ABCD using scale a factor of 2 and center of dilation at (–3, 1), then translate it 4 units to the right.

• not similar; In rectangle ABCD, side AB has a length of 2 units.  The corresponding side in rectangle EFGH has a length of only 1 unit.

• not similar; The ratios of the side lengths are not the same for sides AB and AD.

Answer explanation

Determine if a dilation occurred by examining the ratios of the side lengths. If the ratios are equal, then the figures are similar. If they are not equal, the figures are not similar.

Are the two figures similar? If so, describe a sequence of transformations that maps one figure onto the other figure. If not, explain why they are not similar.

• similar; Dilate triangle ABC using a scale factor of 7676 and center of dilation at the origin, then reflect it over the x-axis.

• similar; Reflect triangle ABC over the x-axis and translate it down 1 and left 1. Then dilate it using a scale factor of 7676 and center of dilation at vertex B.

• not similar; The longest side in triangle ABC is 3.6 unit, but the longest side in triangle DEF is 4.2 units. Corresponding sides are not equal.

Answer explanation

Determine if a dilation occurred by examining the ratios of the side lengths.

Triangle ABC is similar to ΔXYZ. Determine which sequence of transformations maps ΔABC onto ΔXYZ.

Dilate triangle ABC using a scale factor of ​ ​ (a)   and center of dilation at the origin. Then ​ (b)   the resulting image ​ (c)  

• 2
• rotate
• 90°  counterclockwise around the origin
• 4
• 0.5
• 0.25
• reflect
• translate
• 1 unit right and 3 units down
• across the x-axis

Answer explanation

Which sequence of transformations can be used to show that two figures are similar but not necessarily congruent?

• dilation and rotation

• reflection and rotation

• rotation and translation

• translation and reflection

Answer explanation

Translations, reflections, and rotations do not change the size or shape of the figure.

A graphic designer enlarges a rectangular image with a length of 3 inches and width of 5 inches by a scale factor of 2. Then he decides that the enlarged image is too large and reduces it by a scale factor of 0.25. Will the final image fit into a rectangular space that has an area of 3.5 square inches? Justify your response.

 

The final image​ ​ (a)   fit into the space because it has an area of ​ (b)   square inches.

• will not
• 3.75
• will
• 3.5
• 3
• 2.75

Answer explanation

Multiply the lengths of the sides by the scale factors to find the new dimensions.

Square ABCD is similar to square EFGH because a dilation with a scale factor of 2 with the center of dilation at the origin, followed by a translation 5 units to the right maps square ABCD onto square EFGH.

 If you perform the translation first and then the dilation, will the squares still map onto one another? Explain.

 

Choose the correct answer and justification.

• yes; Order does not matter when performing transformations. Square ABCD is still dilated by a factor of 2 and translated to the right 5 units, so it will coincide with square EFGH.

• yes; The ratio between the corresponding sides of square ABCD and square EFGH remains 1 : 2 regardless of the order in which the transformations are performed.  

• no; If the translation of 5 units to the right is performed first, the center of dilation will also need to be moved 5 units to the right.

• no; If the translation is performed first, you only need to dilate square ABCD by a scale factor of 7676 to make it coincide with square EFGH.

Find the Error  A student concluded that rectangle ABCD is similar to rectangle EFGH because a dilation with a scale factor of 0.5 and a translation maps rectangle ABCD onto rectangle EFGH. Find the student's mistake and correct it.

• The student did not state the center of dilation. The student should have said that rectangle ABCD is similar to rectangle EFGH because a dilation with center at the origin and a translation 4 units to the left maps it onto rectangle EFGH.

• The student should have concluded that rectangle ABCD and rectangle EFGH are not similar because not all of the corresponding sides are congruent. Corresponding sides AD and EH have different lengths. 

• The student should have concluded that that rectangle ABCD and rectangle EFGH are not similar because their corresponding sides are not proportional. For example, the ratio AB : EF = 1, but the ratio AD : EH = 2.

• The student did not correctly apply the definition of similar. When two figures are similar, they have the same shape. The student should have said that figures ABCD and EFGH are similar because they are both rectangles. 

Answer explanation

Compare the ratios of the sides of the figures.