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Geometry - Coordinates in all four quadrants

Grade 6Cambridge (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 vertical yy-axis. The origin (0,0)(0, 0) is where the two axes intersect.

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Any point in the plane is represented as an ordered pair (x,y)(x, y). The first number (xx-coordinate) tells you how far to move left or right, and the second number (yy-coordinate) tells you how far to move up or down.

Diagram showing point P at (3, 4) with lines indicating movements from the origin.
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Horizontal distance between two points with the same yy-coordinate is found using ∣x2−x1∣|x_2 - x_1|. Similarly, vertical distance is found using ∣y2−y1∣|y_2 - y_1| when xx-coordinates are the same.

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Geometric shapes can be plotted on the grid. Properties of shapes (like equal side lengths in a square) help determine missing coordinates.

📐Formulae

Horizontal Distance=∣x2−x1∣ (when y-coordinates are the same)\text{Horizontal Distance} = |x_2 - x_1| \text{ (when y-coordinates are the same)}

Vertical Distance=∣y2−y1∣ (when x-coordinates are the same)\text{Vertical Distance} = |y_2 - y_1| \text{ (when x-coordinates are the same)}

Midpoint=(x1+x22,y1+y22)\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

💡Examples

Problem 1:

Identify which quadrant the point P(−4,−2)P(-4, -2) lies in.

Solution:

Quadrant III

Explanation:

In Quadrant III, both the x-coordinate and the y-coordinate are negative. Since -4 and -2 are both negative, the point is in the third quadrant.

Problem 2:

Find the distance between the points A(−3,5)A(-3, 5) and B(4,5)B(4, 5).

Solution:

7 units

Explanation:

Since the y-coordinates are the same (5), the points lie on a horizontal line. The distance is the absolute difference between the x-coordinates: ∣4−(−3)∣=∣4+3∣=7|4 - (-3)| = |4 + 3| = 7.

Problem 3:

Three vertices of a rectangle are (1,2)(1, 2), (5,2)(5, 2), and (5,−3)(5, -3). Find the coordinates of the fourth vertex.

Solution:

(1,−3)(1, -3)

Explanation:

In a rectangle, sides are parallel to the axes. The point (1,2)(1, 2) and (5,2)(5, 2) form the top side. The point (5,2)(5, 2) and (5,−3)(5, -3) form the right side. To complete the rectangle, the fourth point must share the x-coordinate of the first point (1) and the y-coordinate of the third point (-3).

Problem 4:

Plot the points A(−2,3)A(-2, 3), B(2,3)B(2, 3), and C(2,−1)C(2, -1). If these are three vertices of a square ABCDABCD, find the coordinates of vertex DD.

A square ABCD plotted on a coordinate grid.

Solution:

The coordinates of vertex DD are (−2,−1)(-2, -1).

Explanation:

To form a square, the sides must be equal and perpendicular. Side ABAB is horizontal with length ∣2−(−2)∣=4|2 - (-2)| = 4. Side BCBC is vertical with length ∣3−(−1)∣=4|3 - (-1)| = 4. To complete the square, vertex DD must be 44 units directly below A(−2,3)A(-2, 3) or 44 units to the left of C(2,−1)C(2, -1), which leads to (−2,−1)(-2, -1).

Problem 5:

Point MM is the midpoint of the line segment joining X(−4,2)X(-4, 2) and Y(2,−2)Y(2, -2). Calculate the coordinates of MM.

A line segment XY with midpoint M on a coordinate plane.

Solution:

M=(−1,0)M = (-1, 0)

Explanation:

Use the midpoint formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). For xx: −4+22=−22=−1\frac{-4 + 2}{2} = \frac{-2}{2} = -1. For yy: 2+(−2)2=02=0\frac{2 + (-2)}{2} = \frac{0}{2} = 0.