Question

question 36 of 39
which sequence of transformations produces an image that is not congruent to the original figure?
a. a reflection across the x - axis followed by a rotation of 180° counterclockwise
b. a translation of 6 units to the left followed by a reflection across the x - axis
c. a translation of 4 units to the left followed by a dilation of a factor of 3
d. a rotation of 90° clockwise followed by a translation of 4 units to the left

Explanation:

Step1: Recall congruence - preserving transformations

Rigid transformations (reflection, rotation, translation) preserve congruence. They only change the position or orientation of a figure, not its size.

Step2: Analyze dilation

Dilation is a non - rigid transformation. A dilation with a factor \(k
eq1\) changes the size of a figure. If the original figure has side lengths \(l_1,l_2,\cdots\), after a dilation with factor \(k = 3\), the new side lengths are \(l_1^\prime=3l_1\), \(l_2^\prime = 3l_2,\cdots\).

Step3: Check each option

• Option A: Reflection (across \(x\) - axis) and rotation (\(180^{\circ}\) counterclockwise) are rigid transformations. The image is congruent to the original.
• Option B: Translation (6 units to the left) and reflection (across \(x\) - axis) are rigid transformations. The image is congruent to the original.
• Option C: Translation (4 units to the left) is a rigid transformation, but dilation (factor of 3) changes the size. The image is not congruent to the original.
• Option D: Rotation (\(90^{\circ}\) clockwise) and translation (4 units to the left) are rigid transformations. The image is congruent to the original.

Answer:

C. A translation of 4 units to the left followed by a dilation of a factor of 3