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Geometry - Coordinates in all four quadrants

Grade 5Cambridge (IGCSE)

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

🔑Concepts

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The coordinate plane is divided into four regions called quadrants by the horizontal xx-axis and the vertical yy-axis. The point where they intersect is the origin (0,0)(0, 0). Each point is described by an ordered pair (x,y)(x, y). In Quadrant I, both coordinates are positive; in Quadrant II, xx is negative and yy is positive; in Quadrant III, both are negative; and in Quadrant IV, xx is positive and yy is negative.

A coordinate plane showing the four quadrants and the origin.
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Plotting points involves moving horizontally from the origin according to the xx-coordinate (right for positive, left for negative) and then vertically according to the yy-coordinate (up for positive, down for negative).

Illustration of plotting point (3, -2) by moving 3 units right and 2 units down.
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Geometric shapes can be represented on the coordinate plane by connecting vertices. For instance, a rectangle can be defined by four points where opposite sides are parallel to the axes.

A rectangle drawn on a coordinate grid with vertices in multiple quadrants.
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Reflections across the axes change the sign of the coordinates. Reflecting (x,y)(x, y) across the xx-axis results in (x,−y)(x, -y). Reflecting across the yy-axis results in (−x,y)(-x, y).

📐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).

Problem 4:

Point MM is located at (−2,−1)(-2, -1). Reflect this point across the yy-axis to get point M′M'. Then, reflect M′M' across the xx-axis to get point M′′M''. State the coordinates of M′M' and M′′M''.

Points M, M prime, and M double prime shown on a coordinate plane.

Solution:

M′=(2,−1)M' = (2, -1) M′′=(2,1)M'' = (2, 1)

Explanation:

To reflect across the yy-axis, change the sign of the xx-coordinate: (−2,−1)→(2,−1)(-2, -1) \rightarrow (2, -1). To reflect across the xx-axis, change the sign of the yy-coordinate: (2,−1)→(2,1)(2, -1) \rightarrow (2, 1).

Problem 5:

A right-angled triangle has vertices at X(−3,2)X(-3, 2), Y(1,2)Y(1, 2), and Z(1,−2)Z(1, -2). Calculate the lengths of the horizontal and vertical sides.

A right-angled triangle XYZ plotted on a coordinate grid.

Solution:

XY=∣1−(−3)∣=4 unitsXY = |1 - (-3)| = 4 \text{ units} YZ=∣2−(−2)∣=4 unitsYZ = |2 - (-2)| = 4 \text{ units}

Explanation:

The side XYXY is horizontal because the yy-coordinates are the same; its length is the difference in xx-values. The side YZYZ is vertical because the xx-coordinates are the same; its length is the difference in yy-values.