Question

consider the congruent triangles shown below
which series of transformations can be used to prove that triangle abc is congruent to triangle abc?
a. a translation of triangle abc 2 units to the right, followed by a reflection across the y - axis
b. a translation of triangle abc 2 units to the right, followed by a rotation of 180° about the origin
c. a reflection of triangle abc across the y - axis, followed by a translation of 2 units to the right
d. a rotation of triangle abc 180° about the origin, followed by a reflection across the x - axis

Explanation:

Step1: Analyze Option A

A translation 2 units right moves \(ABC\) towards \(y -\)axis. Then reflection across \(y -\)axis would not map \(ABC\) to \(A'B'C'\).

Step2: Analyze Option B

Translation 2 units right and then \(180^{\circ}\) rotation about origin would not map \(ABC\) to \(A'B'C'\) as per the coordinate positions.

Step3: Analyze Option C

Reflection of \(ABC\) across \(y -\)axis flips it over \(y -\)axis. Then translation of 2 units to the right maps \(ABC\) to \(A'B'C'\) correctly.

Step4: Analyze Option D

\(180^{\circ}\) rotation about origin and then reflection across \(x -\)axis would not give the correct mapping of \(ABC\) to \(A'B'C'\) as per the coordinate positions.

Answer:

C. a reflection of triangle \(ABC\) across the \(y -\)axis, followed by a translation of 2 units to the right