Transformations and Symmetry - ISEE Middle Level Quantitative Reasoning

1. Identify the type of symmetry: a figure can be mapped onto itself by a rotation about a point.

   Rotational symmetry. This symmetry allows the figure to coincide with itself after rotation by a specific angle around a center point.

2. What is the transformation rule for reflecting a point $(x,y)$ across the line $y=-x$?

   $(x,y)\rightarrow(-y,-x)$. Reflection across the line $y=-x$ swaps the coordinates and negates both to mirror the point over the anti-diagonal.

3. A segment of length $8$ is dilated by scale factor $\frac{3}{2}$. What is the new length?

   $12$. The new length is the original length multiplied by the scale factor $\frac{3}{2}$.

4. What is the transformation rule for reflecting a point $(x,y)$ across the $x$-axis?

   $(x,y)\rightarrow(x,-y)$. Reflection across the x-axis negates the y-coordinate while preserving the x-coordinate to mirror the point below or above the axis.

5. What is the transformation rule for translating a point $(x,y)$ by $(a,b)$?

   $(x,y)\rightarrow(x+a,y+b)$. Translation shifts the point by adding a to the x-coordinate and b to the y-coordinate.

6. What is the transformation rule for a $90^\circ$ clockwise rotation about the origin?

   $(x,y)\rightarrow(y,-x)$. A $90^\circ$ clockwise rotation transforms the coordinates by setting new x to y and new y to -x.

7. What is the transformation rule for reflecting a point $(x,y)$ across the $y$-axis?

   $(x,y)\rightarrow(-x,y)$. Reflection across the y-axis negates the x-coordinate while preserving the y-coordinate to mirror the point left or right of the axis.

8. Find the image of point $B(-4,2)$ after reflection across the $y$-axis.

   $(4,2)$. Apply the y-axis reflection rule by negating the x-coordinate of (-4,2).

9. Find the image of point $F(1,-3)$ after translation by $(4,-2)$.

   $(5,-5)$. Apply the translation by adding 4 to x and -2 to y of (1,-3).

10. What happens to area under a dilation with scale factor $k$ (in terms of the original area $A$)?

    New area $=k^2A$. Areas scale by the square of the linear scale factor under dilation.

11. A rectangle has area $10$. It is dilated by scale factor $3$. What is the new area?

    $90$. The new area is the original area multiplied by the square of the scale factor 3.

12. What is the transformation rule for dilating a point $(x,y)$ by scale factor $k$ about the origin?

    $(x,y)\rightarrow(kx,ky)$. Dilation enlarges or reduces the figure by multiplying both coordinates by the scale factor k from the origin.

13. Find the image of point $E(-6,1)$ after a $180^\circ$ rotation about the origin.

    $(6,-1)$. Apply the $180^\circ$ rotation rule to (-6,1) by negating both coordinates.

14. What is the transformation rule for a $90^\circ$ counterclockwise rotation about the origin?

    $(x,y)\rightarrow(-y,x)$. A $90^\circ$ counterclockwise rotation transforms the coordinates by setting new x to -y and new y to x.

15. Which transformations always preserve distance and angle measure: translation, rotation, reflection, or dilation?

    Translation, rotation, and reflection. These are rigid motions that preserve congruence, maintaining distances and angles, unlike dilation which alters size.

16. Find the image of point $D(2,9)$ after a $90^\circ$ counterclockwise rotation about the origin.

    $(-9,2)$. Apply the $90^\circ$ counterclockwise rotation rule to (2,9) by setting new x to -9 and new y to 2.

17. Identify the type of symmetry: a figure can be mapped onto itself by a reflection across a line.

    Line symmetry (reflectional symmetry). This symmetry allows the figure to coincide with itself when folded along the line of reflection.

18. What is the smallest positive rotation angle if a figure has rotational symmetry of order $5$?

    $72^\circ$. The smallest angle is $360^\circ$ divided by the order of rotational symmetry.

19. What is the transformation rule for reflecting a point $(x,y)$ across the line $y=x$?

    $(x,y)\rightarrow(y,x)$. Reflection across the line $y=x$ swaps the x- and y-coordinates to mirror the point over the diagonal.

20. What is the transformation rule for a $180^\circ$ rotation about the origin?

    $(x,y)\rightarrow(-x,-y)$. A $180^\circ$ rotation negates both x- and y-coordinates to rotate the point halfway around the origin.

21. Find the image of point $A(3,-5)$ after reflection across the $x$-axis.

    $(3,5)$. Apply the x-axis reflection rule by negating the y-coordinate of (3,-5).

22. Find the image of point $C(7,-1)$ after reflection across the line $y=x$.

    $(-1,7)$. Apply the $y=x$ reflection rule by swapping the coordinates of (7,-1).

23. What is the scale factor $k$ for a dilation if a segment of length $L$ becomes length $L'$?

    $k=\frac{L'}{L}$. The scale factor is the ratio of the image length to the original length under dilation.

24. What is the order of rotational symmetry if the smallest rotation that maps a figure onto itself is $120^\circ$?

    $3$. The order is $360^\circ$ divided by the smallest rotation angle that maps the figure onto itself.

25. A figure has line symmetry about the $y$-axis. If it contains $(5,-2)$, what reflected point must it contain?

    $(-5,-2)$. Symmetry about the y-axis requires the point symmetric to (5,-2) by negating its x-coordinate.

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