Coordinate Geometry Figures - ISEE Lower Level: Quantitative Reasoning
Card 1 of 25
Identify whether $A(0,0),B(4,0),C(3,2),D(-1,2)$ form a trapezoid.
Identify whether $A(0,0),B(4,0),C(3,2),D(-1,2)$ form a trapezoid.
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Yes, it is a trapezoid. At least one pair of opposite sides is parallel, fulfilling the trapezoid definition.
Yes, it is a trapezoid. At least one pair of opposite sides is parallel, fulfilling the trapezoid definition.
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Identify whether $A(0,0),B(3,0),C(4,2),D(1,2)$ form a parallelogram.
Identify whether $A(0,0),B(3,0),C(4,2),D(1,2)$ form a parallelogram.
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Yes, it is a parallelogram. Opposite sides have equal slopes, confirming parallelism for a parallelogram.
Yes, it is a parallelogram. Opposite sides have equal slopes, confirming parallelism for a parallelogram.
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Identify whether $A(0,0),B(2,2),C(4,0),D(2,-2)$ form a square.
Identify whether $A(0,0),B(2,2),C(4,0),D(2,-2)$ form a square.
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Yes, it is a square. The points have equal side lengths and perpendicular adjacent sides, meeting square criteria.
Yes, it is a square. The points have equal side lengths and perpendicular adjacent sides, meeting square criteria.
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Identify whether $A(0,0),B(4,0),C(4,3),D(0,3)$ form a rectangle.
Identify whether $A(0,0),B(4,0),C(4,3),D(0,3)$ form a rectangle.
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Yes, it is a rectangle. The points form right angles with equal opposite sides, satisfying rectangle conditions.
Yes, it is a rectangle. The points form right angles with equal opposite sides, satisfying rectangle conditions.
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What is a quick coordinate test that a quadrilateral is a rectangle using diagonals $AC$ and $BD$?
What is a quick coordinate test that a quadrilateral is a rectangle using diagonals $AC$ and $BD$?
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It is a parallelogram and $AC=BD$. Equal diagonals in a parallelogram imply congruent triangles and right angles, confirming a rectangle.
It is a parallelogram and $AC=BD$. Equal diagonals in a parallelogram imply congruent triangles and right angles, confirming a rectangle.
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Which condition identifies a kite using points $A,B,C,D$ in order?
Which condition identifies a kite using points $A,B,C,D$ in order?
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Two pairs of adjacent equal sides: $AB=AD$ and $BC=CD$. Pairs of equal adjacent sides create the symmetric shape of a kite.
Two pairs of adjacent equal sides: $AB=AD$ and $BC=CD$. Pairs of equal adjacent sides create the symmetric shape of a kite.
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Which condition identifies a trapezoid using points $A,B,C,D$ in order?
Which condition identifies a trapezoid using points $A,B,C,D$ in order?
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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.
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.
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Which condition identifies a rhombus using points $A,B,C,D$ in order?
Which condition identifies a rhombus using points $A,B,C,D$ in order?
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Parallelogram with all sides equal: $AB=BC=CD=DA$. Equal lengths of all sides in a parallelogram characterize a rhombus.
Parallelogram with all sides equal: $AB=BC=CD=DA$. Equal lengths of all sides in a parallelogram characterize a rhombus.
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Which condition identifies a parallelogram using points $A,B,C,D$ in order?
Which condition identifies a parallelogram using points $A,B,C,D$ in order?
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Opposite sides parallel: $m_{AB}=m_{CD}$ and $m_{BC}=m_{AD}$. Matching slopes for opposite sides ensure they are parallel, defining a parallelogram.
Opposite sides parallel: $m_{AB}=m_{CD}$ and $m_{BC}=m_{AD}$. Matching slopes for opposite sides ensure they are parallel, defining a parallelogram.
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Which condition identifies a square using coordinates for consecutive vertices $A,B,C,D$?
Which condition identifies a square using coordinates for consecutive vertices $A,B,C,D$?
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Rectangle condition plus $AB=BC$ (all sides equal). Adding equal side lengths to rectangle properties confirms the figure is a square.
Rectangle condition plus $AB=BC$ (all sides equal). Adding equal side lengths to rectangle properties confirms the figure is a square.
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What coordinate condition guarantees a rectangle is axis-aligned (sides parallel to axes)?
What coordinate condition guarantees a rectangle is axis-aligned (sides parallel to axes)?
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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.
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.
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Identify the slope of a horizontal line such as $y=-2$.
Identify the slope of a horizontal line such as $y=-2$.
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Slope $0$. A horizontal line has constant y and varying x, yielding zero change in y over change in x.
Slope $0$. A horizontal line has constant y and varying x, yielding zero change in y over change in x.
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Identify the slope of a vertical line such as $x=3$.
Identify the slope of a vertical line such as $x=3$.
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Undefined slope. A vertical line has constant x and varying y, resulting in division by zero in the slope formula.
Undefined slope. A vertical line has constant x and varying y, resulting in division by zero in the slope formula.
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What is the condition for two lines to be parallel in the coordinate plane?
What is the condition for two lines to be parallel in the coordinate plane?
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Equal slopes: $m_1=m_2$ (or both vertical). Identical slopes indicate that the lines maintain the same direction and never intersect.
Equal slopes: $m_1=m_2$ (or both vertical). Identical slopes indicate that the lines maintain the same direction and never intersect.
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What is the condition for two nonvertical lines with slopes $m_1$ and $m_2$ to be perpendicular?
What is the condition for two nonvertical lines with slopes $m_1$ and $m_2$ to be perpendicular?
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$m_1m_2=-1$. The product of the slopes equaling $-1$ confirms that the lines intersect at right angles.
$m_1m_2=-1$. The product of the slopes equaling $-1$ confirms that the lines intersect at right angles.
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What is the midpoint formula for the segment with endpoints $P(x_1,y_1)$ and $Q(x_2,y_2)$?
What is the midpoint formula for the segment with endpoints $P(x_1,y_1)$ and $Q(x_2,y_2)$?
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$M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. This formula averages the x- and y-coordinates to locate the center point of the line segment.
$M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. This formula averages the x- and y-coordinates to locate the center point of the line segment.
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What is the distance formula between $P(x_1,y_1)$ and $Q(x_2,y_2)$ on a coordinate plane?
What is the distance formula between $P(x_1,y_1)$ and $Q(x_2,y_2)$ on a coordinate plane?
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$PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. This formula applies the Pythagorean theorem to find the straight-line distance between two points in the plane.
$PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. This formula applies the Pythagorean theorem to find the straight-line distance between two points in the plane.
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What is the slope formula for points $P(x_1,y_1)$ and $Q(x_2,y_2)$?
What is the slope formula for points $P(x_1,y_1)$ and $Q(x_2,y_2)$?
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$m=\frac{y_2-y_1}{x_2-x_1}$. This formula computes the rate of change in the y-direction per unit change in the x-direction between two points.
$m=\frac{y_2-y_1}{x_2-x_1}$. This formula computes the rate of change in the y-direction per unit change in the x-direction between two points.
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What are the necessary and sufficient conditions for a rectangle using points $A,B,C,D$ in order?
What are the necessary and sufficient conditions for a rectangle using points $A,B,C,D$ in order?
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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.
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.
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Identify whether $A(0,0),B(2,2),C(4,4),D(0,4)$ can be a rectangle using all four points as vertices.
Identify whether $A(0,0),B(2,2),C(4,4),D(0,4)$ can be a rectangle using all four points as vertices.
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No, it cannot be a rectangle. Three points are collinear, preventing the formation of a quadrilateral with right angles and equal opposite sides.
No, it cannot be a rectangle. Three points are collinear, preventing the formation of a quadrilateral with right angles and equal opposite sides.
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Identify whether points $A(0,1)$, $B(2,3)$, $C(5,0)$ are collinear.
Identify whether points $A(0,1)$, $B(2,3)$, $C(5,0)$ are collinear.
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No, they are not collinear. Different slopes between AB and BC indicate the points do not lie on a single straight line.
No, they are not collinear. Different slopes between AB and BC indicate the points do not lie on a single straight line.
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Find the area of the axis-aligned rectangle with opposite vertices $(-1,-2)$ and $(4,3)$.
Find the area of the axis-aligned rectangle with opposite vertices $(-1,-2)$ and $(4,3)$.
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Area $25$. The area is the product of the differences in x and y coordinates between opposite corners.
Area $25$. The area is the product of the differences in x and y coordinates between opposite corners.
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What is the missing vertex of an axis-aligned rectangle with vertices $(-2,1)$, $(-2,5)$, and $(3,1)$?
What is the missing vertex of an axis-aligned rectangle with vertices $(-2,1)$, $(-2,5)$, and $(3,1)$?
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$(3,5)$. The missing vertex completes the rectangle by matching the unpaired x and y coordinates.
$(3,5)$. The missing vertex completes the rectangle by matching the unpaired x and y coordinates.
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What is the missing vertex of an axis-aligned rectangle with $A(1,2)$, $B(1,6)$, and $C(5,6)$?
What is the missing vertex of an axis-aligned rectangle with $A(1,2)$, $B(1,6)$, and $C(5,6)$?
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$D(5,2)$. For an axis-aligned rectangle, the missing vertex shares x with C and y with A.
$D(5,2)$. For an axis-aligned rectangle, the missing vertex shares x with C and y with A.
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Identify whether $A(0,0),B(2,0),C(3,1),D(1,1)$ form a rectangle.
Identify whether $A(0,0),B(2,0),C(3,1),D(1,1)$ form a rectangle.
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No, it is not a rectangle. Adjacent sides are not perpendicular, failing the rectangle requirement despite parallelism.
No, it is not a rectangle. Adjacent sides are not perpendicular, failing the rectangle requirement despite parallelism.
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