Question

a. how can you decide if $\triangle def \cong \triangle def$?
b. is $\triangle def \cong \triangle def$? explain.

$\triangle def$ to $\triangle def$.
b. choose the correct answer below.
a. yes, because a rotation of $180^\circ$ about the origin, followed by translations of 3 units down and 5 units left maps $\triangle def$ to $\triangle def$.
b. yes, because a rotation of $180^\circ$ about the origin, followed by translations of 5 units up and 3 units right maps $\triangle def$ to $\triangle def$.
c. yes, because translations of 5 units down and 3 units left, followed by a rotation of $90^\circ$ about the origin and a reflection across the x - axis maps $\triangle def$ to $\triangle def$.
d. no, because a sequence of transformations does not map $\triangle def$ to $\triangle def$.

Explanation:

Step1: Analyze Rotation and Translation

A rotation of \(180^\circ\) about the origin changes \((x,y)\) to \((-x,-y)\). Then, translating 5 units left (subtract 5 from \(x\)) and 3 units down (subtract 3 from \(y\)) should map \(\triangle DEF\) to \(\triangle D'E'F'\). Option A describes this sequence.

Step2: Evaluate Other Options

• Option B: Rotation \(180^\circ\) then translation 5 up and 3 right doesn't match the graph's transformation.
• Option C: Translations 5 down and 3 left then \(90^\circ\) rotation is incorrect.
• Option D: Claiming no mapping is wrong as congruent triangles can be mapped via rigid transformations.

Answer:

A. Yes, because a rotation of \(180^\circ\) about the origin, followed by translations of 3 units down and 5 units left maps \(\triangle DEF\) to \(\triangle D'E'F'\).