Question

lesson 3 9 - 30 performing sequences of rigid transfo perform the given sequence of transformations on each figure. write coordinates of the vertices of the final image. then tell whether the is congruent to the original figure. 1 reflect across the x - axis. translate 5 units left. 2 rotate 90° clockwise reflect across t 3 translate 2 units right and 4 units down. rotate 180° around the origin. 4 reflect acro counterclo

Explanation:

Step1: Recall transformation rules

• Reflection across x - axis: $(x,y)\to(x, - y)$
• Translation: $(x,y)\to(x - 5,y)$ (5 units left)

Let's assume a general point $(x,y)$ on the original figure.

Step2: First transformation (reflection across x - axis)

If a point has coordinates $(x,y)$, after reflection across the x - axis, its new coordinates are $(x,-y)$.

Step3: Second transformation (translation 5 units left)

After the reflection, when we translate 5 units left, the coordinates of the point become $(x - 5,-y)$.
Since rigid transformations (reflection and translation) preserve side - lengths and angles, the final image is congruent to the original figure.

For the third problem:

Step1: Recall transformation rules

Translation: $(x,y)\to(x + 2,y-4)$ (2 units right and 4 units down)
Rotation 180° around the origin: $(x,y)\to(-x,-y)$
Let a point have coordinates $(x,y)$ on the original figure.

Step2: First transformation (translation)

After translating 2 units right and 4 units down, the point's coordinates become $(x + 2,y - 4)$.

Step3: Second transformation (rotation 180° around the origin)

After rotating the point $(x + 2,y - 4)$ 180° around the origin, its new coordinates are $(-(x + 2),-(y - 4))=(-x - 2,-y + 4)$.
Since rigid transformations (translation and rotation) preserve side - lengths and angles, the final image is congruent to the original figure.

Since we don't have the actual coordinates of the vertices of the figures given in the problem (only the general instructions for transformations), we have shown the general process for finding the new coordinates and determining congruence. Rigid transformations (reflections, translations, rotations) always result in a figure that is congruent to the original figure because they preserve distance and angle measures.

Answer:

For the first set of transformations (reflect across x - axis and translate 5 units left): The final image is congruent to the original figure. General new coordinates of a point $(x,y)$ are $(x - 5,-y)$.
For the third set of transformations (translate 2 units right and 4 units down, then rotate 180° around the origin): The final image is congruent to the original figure. General new coordinates of a point $(x,y)$ are $(-x - 2,-y + 4)$.