Question

△abc was transformed into △def. determine which statement about this transformation is correct. statement #1. △abc≅△def because △abc was shifted 2 units to the left and 3 units down, then reflected across the y - axis. statement #2. △abc≅△def because △abc was shifted 2 units to the right and 3 units down, then rotated 180° (counterclockwise). statement #3. △abc≅△def because △abc was shifted 2 units to the right and 3 units down, then rotated - 270° (clockwise)

Explanation:

Step1: Analyze translation

Let's assume a general point \((x,y)\) in \(\triangle ABC\). If we shift a point 2 units to the left and 3 units down, the transformation rule for translation is \((x,y)\to(x - 2,y - 3)\). Then reflecting across the y - axis changes the x - coordinate sign, so \((x-2,y - 3)\to(2 - x,y - 3)\).

Step2: Analyze rotation

For a 180 - degree counter - clockwise rotation about the origin, the transformation rule for a point \((x,y)\) is \((x,y)\to(-x,-y)\). If we first shift 2 units to the right and 3 units down \((x,y)\to(x + 2,y - 3)\), then rotate 180 degrees counter - clockwise, we get \((x + 2,y - 3)\to(-x - 2,-y + 3)\).

Step3: Analyze another rotation

For a - 270 - degree (or 90 - degree counter - clockwise) rotation about the origin, the transformation rule for a point \((x,y)\) is \((x,y)\to(-y,x)\). If we first shift 2 units to the right and 3 units down \((x,y)\to(x + 2,y - 3)\), then rotate 90 degrees counter - clockwise, we get \((x + 2,y - 3)\to(3 - y,x + 2)\).
Let's assume a vertex of \(\triangle ABC\) is \(A(3,3)\).
For statement #1:
Translation: \((3,3)\to(3-2,3 - 3)=(1,0)\), Reflection: \((1,0)\to(-1,0)\)
For statement #2:
Translation: \((3,3)\to(3 + 2,3 - 3)=(5,0)\), Rotation: \((5,0)\to(-5,0)\)
For statement #3:
Translation: \((3,3)\to(3 + 2,3 - 3)=(5,0)\), Rotation: \((5,0)\to(0,5)\)
By observing the graph and applying the transformation rules for congruent triangles, we know that rigid motions (translations, reflections, rotations) preserve congruence.
Let's check the transformation step - by - step. If we shift \(\triangle ABC\) 2 units to the left and 3 units down, and then reflect across the y - axis, we can match the vertices of \(\triangle ABC\) and \(\triangle DEF\).

Answer:

Statement #1 is correct.