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Problem Recap:

The graph displays a rhombus $TUVW$.

1. $TUVW$ is rotated 90° clockwise about the origin to produce $T'U'V'W'$.
2. Then $T'U'V'W'$ is reflected across the y-axis to produce $T''U''V''W''$.

Your task is to verify which statements about the transformations are true.

Steps:

1. Understanding 90° clockwise rotation:

• For a point $(x, y)$:
  • After a 90° clockwise rotation about the origin, the coordinates become $(y, -x)$.

2. Reflecting across the y-axis:

• For a point $(x, y)$:
  • After reflecting across the y-axis, the coordinates become $(-x, y)$.

3. Verifying statements:

Now, I'll analyze the true/false nature of each statement. I'll compute some coordinates after each transformation:

1. Vertices of $TUVW$:

   • $T(-3, 5)$
   • $U(-5, 9)$
   • $V(3, 9)$
   • $W(5, 5)$

2. Vertices of $T'U'V'W'$ after 90° clockwise rotation:

   • $T' = (5, 3)$
   • $U' = (9, 5)$
   • $V' = (9, -3)$
   • $W' = (5, -5)$

3. Vertices of $T''U''V''W''$ after reflection across the y-axis:

   • $T'' = (-5, 3)$
   • $U'' = (-9, 5)$
   • $V'' = (-9, -3)$
   • $W'' = (-5, -5)$

Analyzing the Statements:

1. The x- and y-coordinates of $W''$ have different signs:

   • $W''(-5, -5)$: Both coordinates are negative.
   • This is FALSE.

2. The y-coordinate of $U''$ and the x-coordinate of $W$ are the same:

   • $U''(-9, 5)$: The y-coordinate is $5$.
   • $W(5, 5)$: The x-coordinate is $5$.
   • This is TRUE.

3. The x- and y-coordinates of $T''$ are the same:

   • $T''(-5, 3)$: The x-coordinate is $-5$, and the y-coordinate is $3$.
   • This is FALSE.

4. The x-coordinate of $V''$ is positive:

   • $V''(-9, -3)$: The x-coordinate is $-9$, which is not positive.
   • This is FALSE.

Final Answer:

• The only correct statement is:
  • The y-coordinate of $U''$ and the x-coordinate of $W$ are the same.

Let me know if you'd like a detailed explanation of any part of the solution!

Related Questions:

1. How do you determine the coordinates after a 180° rotation?
2. What happens when reflecting a shape across the x-axis?
3. Can we use matrices to represent transformations like rotations or reflections?
4. What properties of a rhombus remain unchanged after rotation or reflection?
5. How can you verify congruence between the original and transformed shapes?

Tip:

Always break down transformations step by step to avoid confusion and errors.