Rigid transformations and dilations Knowledge-Based Practice

Questions

1. 1.

   A line segment has a slope of $-\frac{3}{4}$. If the segment is reflected across the $x$-axis, translated 5 units to the left, and then dilated by a scale factor of 2 centered at the origin, what is the slope of the resulting image segment?

   • A) $-\frac{3}{2}$
   • B) $-\frac{3}{4}$
   • C) $\frac{3}{2}$
   • D) $\frac{3}{4}$
2. 2.

   A polygon with an area of 24 square centimeters is translated 6 units up, reflected across the $x$-axis, and then dilated by a scale factor of 3. What is the area of the resulting image?

   • A) 72 square centimeters
   • B) 144 square centimeters
   • C) 216 square centimeters
   • D) 648 square centimeters
3. 3.

   A triangle with an interior angle measuring $40^\circ$ and a perimeter of 18 cm is rotated $180^\circ$ about the origin and then dilated by a scale factor of 2.5. What are the measure of the corresponding angle and the perimeter of the resulting triangle?

   • A) $100^\circ$ and 45 cm
   • B) $40^\circ$ and 18 cm
   • C) $40^\circ$ and 45 cm
   • D) $100^\circ$ and 112.5 cm
4. 4.

   A polygon has a vertex at $A(2, 4)$. If the polygon is rotated $90^\circ$ counterclockwise about the origin and then dilated by a scale factor of 3 with the center of dilation at the origin, what are the coordinates of the resulting image vertex $A''$?

   • A) $(-6, 12)$
   • B) $(6, 12)$
   • C) $(-12, -6)$
   • D) $(-12, 6)$
5. 5.

   Triangle $ABC$ undergoes a reflection across the line $y = x$, followed by a translation by $\langle -3, 5 \rangle$, and finally a dilation centered at the origin with a scale factor of $k = \frac{1}{2}$ to produce triangle $A'B'C'$. Which statement correctly describes the geometric relationship between triangle $ABC$ and triangle $A'B'C'$?

   • A) Triangle $A'B'C'$ is congruent to triangle $ABC$ because reflections and translations preserve side lengths.
   • B) Triangle $A'B'C'$ is similar but not congruent to triangle $ABC$ because the sequence includes a dilation with a scale factor other than 1.
   • C) Triangle $A'B'C'$ is neither similar nor congruent to triangle $ABC$ because the reflection alters the orientation of the vertices.
   • D) Triangle $A'B'C'$ is congruent to triangle $ABC$ because a dilation centered at the origin preserves all side lengths.

Answers and step-by-step solutions appear after you finish the test.