Question

triangle flags you are making two triangle flags for a project and need the flags to be the same shape and size. \\(\triangle xyz\\) and \\(\triangle xyz\\) are the flags you have drawn. are the flags the same shape and size? explain.

choose the correct answer below.

a. yes, because reflections across the x - axis, and the y - axis, followed by a translation of 5 units up, map \\(\triangle xyz\\) to \\(\triangle xyz\\).

b. yes, because translations of 5 units up and 3 units left, followed by a reflection across the x - axis, maps \\(\triangle xyz\\) to \\(\triangle xyz\\).

c. yes, because a reflection across the y - axis, followed by a translation of 5 units down and 3 units right, maps \\(\triangle xyz\\) to \\(\triangle xyz\\).

d. no, because a sequence of rigid motions does not map \\(\triangle xyz\\) to \\(\triangle xyz\\).

Explanation:

Step1: Analyze Rigid Motions

Rigid motions (reflections, translations) preserve shape and size. Check transformations between \( \triangle XYZ \) and \( \triangle X'Y'Z' \).

Step2: Identify Transformations

First, reflect \( \triangle XYZ \) across the \( y \)-axis (flips over \( y \)-axis). Then translate: 5 units down (vertical) and 3 units right (horizontal). These are rigid motions, so shape/size remain.

Step3: Evaluate Options

• Option A: Reflection over \( x \)-axis is incorrect (visual check shows \( y \)-axis reflection).
• Option B: Translation direction (up/left) is incorrect.
• Option C: Matches reflection over \( y \)-axis, then translation 5 down, 3 right.
• Option D: Incorrect, as rigid motions do preserve shape/size.

Answer:

C. Yes, because a reflection across the y - axis, followed by a translation of 5 units down and 3 units right, maps \( \triangle XYZ \) to \( \triangle X'Y'Z' \).