Question

select the correct answer.

shape 1 and shape 2 are plotted on a coordinate plane. which rigid transformation can you perform on shape 2 to show that shape 2 is congruent to shape 1?

a 45° rotation clockwise about the origin
a translation 6 units to the left and 8 units up
a reflection across the x-axis
a translation 3 units to the left and 6 units up

Explanation:

Identify corresponding vertices on both shapes

Using the Rigid Transformations knowledge point
Let us identify a distinct vertex on shape 2 and find its corresponding vertex on shape 1.

• The bottom-right corner of shape 2 is at \((12, -2)\).
• The bottom-right corner of shape 1 is at \((6, 6)\).

Calculate the horizontal translation

Using the Translation of Figures knowledge point

$$ \Delta x = x_{\text{target}} - x_{\text{source}} = 6 - 12 = -6 $$

This represents a translation of 6 units to the left.

Calculate the vertical translation

Using the Translation of Figures knowledge point

$$ \Delta y = y_{\text{target}} - y_{\text{source}} = 6 - (-2) = 8 $$

This represents a translation of 8 units up.

Verify with another vertex

Using the Rigid Transformations knowledge point
Let us test the top-most vertex:

• Top-most vertex of shape 2 is at \((10, 6)\).
• Top-most vertex of shape 1 is at \((4, 14)\).
• Applying the translation:

$$ (10 - 6, 6 + 8) = (4, 14) $$

The translation is consistent for all vertices.

Answer:

• (A) a 45° rotation clockwise about the origin
• (B) a translation 6 units to the left and 8 units up (Correct answer)
• (C) a reflection across the x-axis
• (D) a translation 3 units to the left and 6 units up