Rigid transformations and dilations Knowledge-Based Download PDF

Questions

1. 1.

   A line with the equation $y = 3x - 4$ is dilated by a scale factor of $2$ with the origin as the center of dilation, and then translated $3$ units up. What is the equation of the resulting line?

   • A) $y = 3x - 11$
   • B) $y = 3x - 5$
   • C) $y = 6x - 5$
   • D) $y = 3x - 1$
2. 2.

   In the coordinate plane, line segment AB has endpoints $A(1, 2)$ and $B(4, 6)$. The segment is reflected across the $y$-axis, translated 5 units down, and then dilated by a scale factor of 2 centered at the origin. What is the length of the resulting image segment?

   • A) 5
   • B) 8
   • C) 10
   • D) 20
3. 3.

   Point $A(-4, 6)$ is rotated $180^\circ$ about the origin, translated according to the rule $(x, y) \to (x - 2, y + 3)$, and then dilated by a scale factor of $\frac{3}{2}$ centered at the origin. What are the coordinates of the resulting point $A'''$?

   • A) $\left(-3, \frac{9}{2}\right)$
   • B) $(4, -6)$
   • C) $(6, -9)$
   • D) $\left(3, -\frac{9}{2}\right)$
4. 4.

   A polygon with an area of 18 square units is translated 4 units left, rotated $90^\circ$ clockwise about the origin, and then dilated by a scale factor of $\frac{1}{3}$. What is the area of the resulting figure?

   • A) 6 square units
   • B) 2 square units
   • C) 54 square units
   • D) 18 square units
5. 5.

   Triangle $ABC$ is reflected across the line $y = x$ and then dilated by a scale factor of 3 centered at the origin to produce triangle $A''B''C''$. Which statement correctly describes the relationship between triangle $ABC$ and triangle $A''B''C''$?

   • A) Triangle $ABC$ and triangle $A''B''C''$ are congruent because reflections preserve both side lengths and angle measures.
   • B) Triangle $ABC$ and triangle $A''B''C''$ are similar but not congruent because the dilation alters side lengths by a factor of 3 while preserving angle measures.
   • C) Triangle $ABC$ and triangle $A''B''C''$ are congruent because a dilation centered at the origin does not change the dimensions of a polygon.
   • D) Triangle $ABC$ and triangle $A''B''C''$ are neither similar nor congruent because performing a reflection followed by a dilation distorts interior angle measures.