Coordinate Geometry Figures - ISEE Lower Level Quantitative Reasoning

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

    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.

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

    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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    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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