Coordinate Shape Analysis - ISEE Middle Level Mathematics Achievement

1. Identify the coordinates after a $90^\circ$ counterclockwise rotation of $(x,y)$ about the origin.

   $(-y,\,x)$. Transforms coordinates by rotating 90 degrees counterclockwise around the origin.

2. Identify the coordinates after a $180^\circ$ rotation of $(x,y)$ about the origin.

   $(-x,\,-y)$. Negates both coordinates to achieve a 180-degree rotation about the origin.

3. What is the midpoint formula for the segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

   $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Calculates the average of the x-coordinates and y-coordinates to find the midpoint of a line segment.

4. Identify the coordinates after reflecting point $(x,y)$ across the $y$-axis.

   $(-x,\,y)$. Negates the x-coordinate to mirror the point over the y-axis.

5. What is the midpoint of the segment with endpoints $(1,5)$ and $(7,1)$?

   $(4,\,3)$. Averages the x-coordinates $(1+7)/2=4$ and y-coordinates $(5+1)/2=3$.

6. What is the slope of a horizontal line segment (in coordinate geometry)?

   $m=0$. No change in y-coordinates results in zero rise over any run for horizontal lines.

7. What is the perimeter of an axis-aligned rectangle with opposite vertices $(x_1,y_1)$ and $(x_2,y_2)$?

   $2\left(|x_2-x_1|+|y_2-y_1|\right)$. Sums twice the absolute differences in x and y for the total length of all sides.

8. What is the slope of the line through points $(2,1)$ and $(5,7)$?

   $2$. Computes rise $(7-1)=6$ over run $(5-2)=3$, simplifying to $2$.

9. What is the area of an axis-aligned rectangle with opposite vertices $(x_1,y_1)$ and $(x_2,y_2)$?

   $|x_2-x_1|\cdot|y_2-y_1|$. Multiplies the absolute differences in x and y coordinates for length and width of the rectangle.

10. What condition on slopes shows two lines are perpendicular on a coordinate plane?

    Negative reciprocals: $m_1\cdot m_2=-1$. Perpendicular lines have slopes whose product is $-1$, indicating a right angle between them.

11. What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$ on a coordinate plane?

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Derived from the Pythagorean theorem applied to the differences in x and y coordinates between two points.

12. What is the area of the axis-aligned rectangle with vertices $(1,2)$, $(5,2)$, $(5,6)$, $(1,6)$?

    $16$. Multiplies width 4 by height 4 for the area of the axis-aligned rectangle.

13. Identify the coordinates after translating point $(x,y)$ by $\langle a,b\rangle$.

    $(x+a,\,y+b)$. Adds the translation vector components to the original coordinates for the new position.

14. Identify the coordinates after reflecting point $(x,y)$ across the $x$-axis.

    $(x,\,-y)$. Negates the y-coordinate to mirror the point over the x-axis.

15. Identify the slope of a line perpendicular to a line with slope $m$ (for $m\neq 0$).

    $-\frac{1}{m}$. The negative reciprocal ensures the lines form a right angle when neither is vertical.

16. What condition on slopes shows two nonvertical lines are parallel on a coordinate plane?

    Equal slopes: $m_1=m_2$. Parallel lines maintain the same steepness, hence identical slopes for nonvertical cases.

17. What is the slope of a vertical line segment (in coordinate geometry)?

    Undefined slope. Division by zero occurs due to no change in x-coordinates for vertical lines.

18. Identify the coordinates after a dilation of $(x,y)$ about the origin with scale factor $k$.

    $(kx,\,ky)$. Multiplies both coordinates by the scale factor to enlarge or reduce from the origin.

19. Identify the coordinates after reflecting point $(x,y)$ across the line $y=x$.

    $(y,\,x)$. Swaps x and y coordinates to reflect over the line of symmetry $y=x$.

20. What is the distance between points $(2,3)$ and $(6,3)$?

    $4$. Applies the distance formula with $Delta x=4$ and $Delta y=0$, yielding $sqrt{16}$.

21. Which option best classifies $AB$ and $CD$ as parallel if $A(0,0)$, $B(2,2)$, $C(0,1)$, $D(2,3)$?

    Parallel, since both slopes are $1$. Both segments have slope $1$, confirming they are parallel lines.

22. What is the slope formula for the line through points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Represents the ratio of the change in y to the change in x, indicating the steepness of the line.

23. What is the area of the triangle with vertices $(0,0)$, $(4,0)$, and $(0,3)$?

    $6$. Uses base 4 and height 3 in the triangle area formula, yielding half of 12.

24. What is the area of a triangle with vertices $(0,0)$, $(b,0)$, and $(0,h)$?

    $\frac{1}{2}bh$. Applies the triangle area formula with base $b$ along the x-axis and height $h$ along the y-axis.

25. Identify whether $AB$ is perpendicular to $BC$ if $A(0,0)$, $B(2,0)$, and $C(2,3)$.

    Perpendicular. Horizontal AB (slope 0) and vertical BC (undefined slope) form a right angle.

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