ISEE Middle Level Mathematics Achievement Flashcards: Coordinate Shape Analysis

What this deck covers

This deck focuses on Coordinate Shape Analysis, giving you a quick way to review the definitions, rules, and examples that matter most for ISEE Middle Level Mathematics Achievement.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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