Question

the graph shows quadrilaterals q and r. which statement justifies whether or not the two quadrilaterals are congruent? a. yes. because r is the result of reflecting q across the x - axis. b. yes. because r is the result of reflecting q across the line y = x. c. no. because r can only be obtained from q by a series of rigid transformations. d. no. because r cannot be obtained from q by a series of rotations and translations. e. yes. because r is the result of rotating q by 90° counter - clockwise about the origin.

Explanation:

Step1: Recall congruence and transformation rules

Two figures are congruent if one can be obtained from the other by a series of rigid - transformations (reflections, rotations, translations). Rigid transformations preserve side - lengths and angles.

Step2: Analyze the transformation from Q to R

By observing the graph, we can see that quadrilateral R is the result of reflecting quadrilateral Q across the x - axis. When a figure is reflected across the x - axis, the x - coordinates of its vertices remain the same and the y - coordinates change their signs. Reflection is a rigid transformation.

Step3: Evaluate each option

• Option A: Correct. Since R is the result of reflecting Q across the x - axis (a rigid transformation), Q and R are congruent.
• Option B: R is not the result of reflecting Q across the line y = x.
• Option C: R can be obtained from Q by a rigid transformation (reflection), so this option is incorrect.
• Option D: R can be obtained from Q by a reflection (a type of rigid transformation), so this option is incorrect.
• Option E: R is not the result of rotating Q by 90° counter - clockwise about the origin.

Answer:

A. Yes, because R is the result of reflecting Q across the x - axis.