ISEE Lower Level Quantitative Reasoning Flashcards: Coordinate Geometry Figures

What this deck covers

This deck focuses on Coordinate Geometry Figures, giving you a quick way to review the definitions, rules, and examples that matter most for ISEE Lower Level Quantitative Reasoning.

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 whether points $A(0,1)$, $B(2,3)$, $C(5,0)$ are collinear.

Answer: No, they are not collinear. Different slopes between AB and BC indicate the points do not lie on a single straight line.

Flashcard 2: Which condition identifies a square using coordinates for consecutive vertices $A,B,C,D$?

Answer: Rectangle condition plus $AB=BC$ (all sides equal). Adding equal side lengths to rectangle properties confirms the figure is a square.

Flashcard 3: What is the slope formula for points $P(x_1,y_1)$ and $Q(x_2,y_2)$?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. This formula computes the rate of change in the y-direction per unit change in the x-direction between two points.

Flashcard 4: What is the missing vertex of an axis-aligned rectangle with $A(1,2)$, $B(1,6)$, and $C(5,6)$?

Answer: $D(5,2)$. For an axis-aligned rectangle, the missing vertex shares x with C and y with A.

Flashcard 5: Identify whether $A(0,0),B(2,2),C(4,4),D(0,4)$ can be a rectangle using all four points as vertices.

Answer: No, it cannot be a rectangle. Three points are collinear, preventing the formation of a quadrilateral with right angles and equal opposite sides.

Flashcard 6: What is the condition for two nonvertical lines with slopes $m_1$ and $m_2$ to be perpendicular?

Answer: $m_1m_2=-1$. The product of the slopes equaling $-1$ confirms that the lines intersect at right angles.

Flashcard 7: What is the condition for two lines to be parallel in the coordinate plane?

Answer: Equal slopes: $m_1=m_2$ (or both vertical). Identical slopes indicate that the lines maintain the same direction and never intersect.

Flashcard 8: What is a quick coordinate test that a quadrilateral is a rectangle using diagonals $AC$ and $BD$?

Answer: It is a parallelogram and $AC=BD$. Equal diagonals in a parallelogram imply congruent triangles and right angles, confirming a rectangle.

Flashcard 9: What is the missing vertex of an axis-aligned rectangle with vertices $(-2,1)$, $(-2,5)$, and $(3,1)$?

Answer: $(3,5)$. The missing vertex completes the rectangle by matching the unpaired x and y coordinates.

Flashcard 10: Identify the slope of a horizontal line such as $y=-2$.

Answer: Slope $0$. A horizontal line has constant y and varying x, yielding zero change in y over change in x.

Flashcard 11: What are the necessary and sufficient conditions for a rectangle using points $A,B,C,D$ in order?

Answer: Adjacent sides perpendicular and opposite sides equal: $AB \perp BC$ and $AB=CD$, $BC=AD$. These conditions ensure all angles are right angles and opposite sides are congruent, defining a rectangle in the coordinate plane.

Flashcard 12: What coordinate condition guarantees a rectangle is axis-aligned (sides parallel to axes)?

Answer: Two distinct $x$-values and two distinct $y$-values among the four vertices. Sharing exactly two x-coordinates and two y-coordinates ensures sides are horizontal and vertical.

Flashcard 13: Which condition identifies a parallelogram using points $A,B,C,D$ in order?

Answer: Opposite sides parallel: $m_{AB}=m_{CD}$ and $m_{BC}=m_{AD}$. Matching slopes for opposite sides ensure they are parallel, defining a parallelogram.

Flashcard 14: Identify whether $A(0,0),B(4,0),C(4,3),D(0,3)$ form a rectangle.

Answer: Yes, it is a rectangle. The points form right angles with equal opposite sides, satisfying rectangle conditions.

Flashcard 15: Identify whether $A(0,0),B(2,2),C(4,0),D(2,-2)$ form a square.

Answer: Yes, it is a square. The points have equal side lengths and perpendicular adjacent sides, meeting square criteria.

Flashcard 16: Which condition identifies a rhombus using points $A,B,C,D$ in order?

Answer: Parallelogram with all sides equal: $AB=BC=CD=DA$. Equal lengths of all sides in a parallelogram characterize a rhombus.

Flashcard 17: Identify whether $A(0,0),B(2,0),C(3,1),D(1,1)$ form a rectangle.

Answer: No, it is not a rectangle. Adjacent sides are not perpendicular, failing the rectangle requirement despite parallelism.

Flashcard 18: What is the distance formula between $P(x_1,y_1)$ and $Q(x_2,y_2)$ on a coordinate plane?

Answer: $PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. This formula applies the Pythagorean theorem to find the straight-line distance between two points in the plane.

Flashcard 19: Identify whether $A(0,0),B(3,0),C(4,2),D(1,2)$ form a parallelogram.

Answer: Yes, it is a parallelogram. Opposite sides have equal slopes, confirming parallelism for a parallelogram.

Flashcard 20: Find the area of the axis-aligned rectangle with opposite vertices $(-1,-2)$ and $(4,3)$.

Answer: Area $25$. The area is the product of the differences in x and y coordinates between opposite corners.

Flashcard 21: Identify the slope of a vertical line such as $x=3$.

Answer: Undefined slope. A vertical line has constant x and varying y, resulting in division by zero in the slope formula.

Flashcard 22: Identify whether $A(0,0),B(4,0),C(3,2),D(-1,2)$ form a trapezoid.

Answer: Yes, it is a trapezoid. At least one pair of opposite sides is parallel, fulfilling the trapezoid definition.

Flashcard 23: Which condition identifies a kite using points $A,B,C,D$ in order?

Answer: Two pairs of adjacent equal sides: $AB=AD$ and $BC=CD$. Pairs of equal adjacent sides create the symmetric shape of a kite.

Flashcard 24: What is the midpoint formula for the segment with endpoints $P(x_1,y_1)$ and $Q(x_2,y_2)$?

Answer: $M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. This formula averages the x- and y-coordinates to locate the center point of the line segment.

Flashcard 25: Which condition identifies a trapezoid using points $A,B,C,D$ in order?

Answer: At least one pair of opposite sides parallel, such as $m_{AB}=m_{CD}$. At least one pair of parallel opposite sides defines a trapezoid in the coordinate plane.