Completing Grid Figures - ISEE Lower Level Mathematics Achievement

1. What is the image of $(6,-1)$ after a $180^\circ$ rotation about the origin?

   $(-6,1)$. Applying the $180^\circ$ rotation rule transforms the point to its image.

2. What is the image of $(2,5)$ after a $90^\circ$ counterclockwise rotation about the origin?

   $(-5,2)$. Applying the $90^\circ$ counterclockwise rotation rule transforms the point to its image.

3. Identify the rule for a $90^\circ$ clockwise rotation about the origin of $(x,y)$.

   $(y,-x)$. A $90^\circ$ clockwise rotation about the origin transforms $(x,y)$ to $(y,-x)$.

4. Identify the rule for a $180^\circ$ rotation about the origin of $(x,y)$.

   $(-x,-y)$. A $180^\circ$ rotation about the origin transforms $(x,y)$ to $(-x,-y)$.

5. Identify the point symmetric to $A(2,-1)$ across the center point $M(5,3)$.

   $(8,7)$. The point symmetric across $M$ is calculated as $2M - A$, ensuring $M$ is the midpoint.

6. What is the reflection of the point $(4,-3)$ across the $x$-axis?

   $(4,3)$. Reflection across the x-axis negates the y-coordinate while keeping the x-coordinate unchanged.

7. What is the fourth vertex of a parallelogram with consecutive vertices $A(-2,1)$, $B(1,4)$, $C(5,3)$?

   $D(2,0)$. The fourth vertex of a parallelogram with consecutive vertices $A$, $B$, $C$ is found using $D = A + C - B$.

8. What is the reflection of the point $(-7,2)$ across the $y$-axis?

   $(7,2)$. Reflection across the y-axis negates the x-coordinate while keeping the y-coordinate unchanged.

9. What point completes an axis-aligned rectangle with vertices $(-3,2)$, $(-3,-4)$, and $(5,-4)$?

   $(5,2)$. The fourth vertex of an axis-aligned rectangle shares the x-coordinate of one point and y-coordinate of another to close the shape.

10. Identify the rule for a $90^\circ$ counterclockwise rotation about the origin of $(x,y)$.

    $(-y,x)$. A $90^\circ$ counterclockwise rotation about the origin transforms $(x,y)$ to $(-y,x)$.

11. Identify the translation of $(x,y)$ by $(-5,4)$ in coordinate form.

    $(x-5,y+4)$. Translation by a vector $(h,k)$ adds $h$ to the x-coordinate and $k$ to the y-coordinate.

12. What is the fourth vertex of a parallelogram with $A(1,2)$, $B(5,2)$, and $C(6,6)$ consecutive?

    $D(2,6)$. The fourth vertex of a parallelogram with consecutive vertices $A$, $B$, $C$ is found using $D = A + C - B$.

13. What point completes a square with vertices $(-1,-1)$, $(3,-1)$, and $(3,3)$?

    $(-1,3)$. The fourth vertex completes the square by ensuring all sides are equal and angles are $90^\circ$.

14. What is the reflection of the point $(3,-8)$ across the origin?

    $(-3,8)$. Reflection across the origin negates both the x-coordinate and y-coordinate.

15. Identify the coordinate formula for the fourth parallelogram vertex $D$ when $A,B,C$ are consecutive.

    $D=A+C-B$. For consecutive vertices $A$, $B$, $C$ in a parallelogram, the vector formula $D = A + C - B$ ensures parallel and equal sides.

16. What point completes a square with vertices $(0,0)$, $(2,0)$, and $(2,2)$?

    $(0,2)$. The fourth vertex completes the square by maintaining equal side lengths and right angles with the given points.

17. What is the slope of the line through $(1,2)$ and $(5,10)$?

    $2$. Slope is calculated as the change in y divided by the change in x between the two points.

18. What is the midpoint of the segment with endpoints $(-2,5)$ and $(6,1)$?

    $(2,3)$. The midpoint is calculated as the average of the x-coordinates and y-coordinates of the endpoints.

19. What point completes a rectangle with vertices $A(1,1)$, $B(1,5)$, and $C(6,5)$?

    $(6,1)$. The fourth vertex of the rectangle shares the x-coordinate with $C$ and the y-coordinate with $A$.

20. What point is $3$ units right and $2$ units down from $(-1,6)$?

    $(2,4)$. Translating right adds to the x-coordinate, and down subtracts from the y-coordinate.

21. What is the image of $(-3,4)$ after a $90^\circ$ clockwise rotation about the origin?

    $(4,3)$. Applying the $90^\circ$ clockwise rotation rule transforms the point to its image.

22. Identify the formula for the point $B(x_2,y_2)$ if $M(x_m,y_m)$ is the midpoint of $AB$ and $A(x_1,y_1)$ is known.

    $B(2x_m-x_1,2y_m-y_1)$. Rearranging the midpoint formula solves for $B$ when $M$ is the midpoint of $A$ and $B$.

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