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Geometry - Coordinates in all four quadrants

Grade 5IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Coordinate Plane: A two-dimensional surface formed by the intersection of a horizontal line (x-axis) and a vertical line (y-axis).

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Origin: The point where the x-axis and y-axis intersect, represented by the coordinates (0, 0).

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Quadrants: The four regions of the coordinate plane. Quadrant I (+,+), Quadrant II (-,+), Quadrant III (-,-), and Quadrant IV (+,-).

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Ordered Pairs: Coordinates are always written in the form (x, y), where x is the horizontal position and y is the vertical position.

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Plotting Points: To plot a point, start at the origin, move along the x-axis first (left for negative, right for positive), then move parallel to the y-axis (down for negative, up for positive).

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Vertex/Vertices: The corners of geometric shapes plotted on a coordinate grid.

📐Formulae

Ordered Pair: (x,y)(x, y)

Midpoint of a line segment: M=(x1+x22,y1+y22)M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})

Translation (Moving a point): (x+a,y+b)(x + a, y + b) where 'a' is horizontal shift and 'b' is vertical shift.

Horizontal Distance between (x1,y)(x_1, y) and (x2,y)(x_2, y): ∣x2−x1∣|x_2 - x_1| (when y-coordinates are the same).

Vertical Distance between (x,y1)(x, y_1) and (x,y2)(x, y_2): ∣y2−y1∣|y_2 - y_1| (when x-coordinates are the same).

💡Examples

Problem 1:

Identify the quadrant in which the point P(−4,3)P(-4, 3) lies.

Solution:

Quadrant II

Explanation:

The x-coordinate is negative (-4) and the y-coordinate is positive (3). In the coordinate plane, negative x and positive y values are located in the top-left section, which is Quadrant II.

Problem 2:

A square has three vertices at A(2,2)A(2, 2), B(−2,2)B(-2, 2), and C(−2,−2)C(-2, -2). Find the coordinates of the fourth vertex DD.

Solution:

D(2,−2)D(2, -2)

Explanation:

In a square, opposite sides are parallel and equal in length. Points A and B are on the line y=2y=2. Points B and C are on the line x=−2x=-2. To complete the square, point D must align horizontally with C (y=−2y=-2) and vertically with A (x=2x=2). Therefore, the point is (2,−2)(2, -2).

Problem 3:

Translate the point K(1,−3)K(1, -3) by 4 units to the left and 2 units up. What are the new coordinates?

Solution:

K′(−3,−1)K'(-3, -1)

Explanation:

Moving 4 units left means subtracting 4 from the x-coordinate: 1−4=−31 - 4 = -3. Moving 2 units up means adding 2 to the y-coordinate: −3+2=−1-3 + 2 = -1. The resulting coordinate is (−3,−1)(-3, -1).