Coordinates

1. Overview

Coordinates are used to pinpoint exact locations on a 2D plane. The Cartesian coordinate system uses an $x$-axis (horizontal) and a $y$-axis (vertical) to define a point's position using an ordered pair of numbers, $(x, y)$. This is crucial for graphing, geometry, and real-world applications like mapping. Mastering coordinates is fundamental to success in IGCSE Mathematics.

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Key Definitions

• Cartesian Plane: A flat, two-dimensional surface defined by a horizontal axis and a vertical axis.
• x-axis: The horizontal number line on a coordinate graph.
• y-axis: The horizontal axis represents the $x$-coordinate.
• y-axis: The vertical number line on a coordinate graph.
• Origin: The point $(0, 0)$ where the x-axis and y-axis intersect.
• Coordinate (Ordered Pair): A pair of numbers $(x, y)$ that describes the position of a point. The order matters!
• Quadrant: One of the four sections of a coordinate plane divided by the axes.

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Core Content

Understanding the Cartesian System

Coordinates are always written as an ordered pair in the form $(x, y)$.

• The first number ($x$) tells you how far to move horizontally from the origin. Positive moves right, negative moves left.
• The second number ($y$) tells you how far to move vertically from the origin. Positive moves up, negative moves down.

Reading Coordinates

To identify the coordinates of a point:

1. Look straight down (or up) to the x-axis to find the $x$-value.
2. Look straight across to the y-axis to find the $y$-value.
3. Write them inside brackets, separated by a comma.

Worked example 1 — Reading a Point

Question: Find the coordinates of point $A$ which is 3 units to the left of the y-axis and 4 units above the x-axis.

• Step 1: Determine the $x$-direction. "3 units left" means $x = -3$.
• Step 2: Determine the $y$-direction. "4 units above" means $y = 4$.
• Step 3: Combine into an ordered pair.
• Answer: $(-3, 4)$

Plotting Coordinates

To plot a point like $(2, -5)$:

1. Start at the origin $(0, 0)$.
2. Move 2 units to the right (positive $x$).
3. Move 5 units down (negative $y$).
4. Mark the point with a small 'x' or a dot.

Worked example 2 — Plotting multiple points

Question: Plot the points $P(0, 3)$, $Q(4, 0)$, and $R(-2, -1)$.

• Point P: $x=0$ (stay on the y-axis), $y=3$ (move up 3).
• Point Q: $x=4$ (move right 4), $y=0$ (stay on the x-axis).
• Point R: $x=-2$ (move left 2), $y=-1$ (move down 1).

Worked example 3 — Finding a missing vertex

Question: Points $A(1, 1)$, $B(1, 5)$, and $C(3, 5)$ are three vertices of a rectangle $ABCD$. Plot the points and state the coordinates of the fourth vertex, $D$.

• Step 1: Plot the points $A$, $B$, and $C$ on a coordinate grid.
• Step 2: Observe the relationship between the points. $AB$ is a vertical line, and $BC$ is a horizontal line. This indicates that $ABCD$ is a rectangle with sides parallel to the axes.
• Step 3: To find the coordinates of $D$, we need to consider that $AD$ must be horizontal and $CD$ must be vertical.
• Step 4: The $x$-coordinate of $D$ must be the same as $C$, which is $3$.
• Step 5: The $y$-coordinate of $D$ must be the same as $A$, which is $1$.
• Step 6: Therefore, the coordinates of $D$ are $(3, 1)$.

Answer: $(3, 1)$

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Extended Content (Extended Only)

While the core concept of coordinates is the same for both Core and Extended students, Extended questions often involve applying coordinate knowledge in more complex geometric problems. This might include finding areas of shapes defined by coordinates, or using coordinates to prove geometric properties.

For example, you might be given the coordinates of three points and asked to show that they form a right-angled triangle. This would involve using the distance formula (related to Pythagoras' theorem) to calculate the lengths of the sides and then verifying that $a^2 + b^2 = c^2$.

Worked example 4 — Showing a right-angled triangle

Question: The points $A(-2, 1)$, $B(2, 3)$, and $C(0, -1)$ are the vertices of a triangle. Show that triangle $ABC$ is a right-angled triangle.

• Step 1: Calculate the length of side $AB$ using the distance formula: $AB = \sqrt{(2 - (-2))^2 + (3 - 1)^2}$ $AB = \sqrt{(4)^2 + (2)^2}$ $AB = \sqrt{16 + 4}$ $AB = \sqrt{20}$
• Step 2: Calculate the length of side $BC$ using the distance formula: $BC = \sqrt{(0 - 2)^2 + (-1 - 3)^2}$ $BC = \sqrt{(-2)^2 + (-4)^2}$ $BC = \sqrt{4 + 16}$ $BC = \sqrt{20}$
• Step 3: Calculate the length of side $AC$ using the distance formula: $AC = \sqrt{(0 - (-2))^2 + (-1 - 1)^2}$ $AC = \sqrt{(2)^2 + (-2)^2}$ $AC = \sqrt{4 + 4}$ $AC = \sqrt{8}$
• Step 4: Check if Pythagoras' theorem holds: $AB^2 + AC^2 = BC^2$ or $AC^2 + BC^2 = AB^2$ or $AB^2 + BC^2 = AC^2$ $(\sqrt{20})^2 = 20$ $(\sqrt{8})^2 = 8$ $(\sqrt{20})^2 = 20$ Check: $8 + 20 = 20$ ? No. Check: $20 + 20 = 8$ ? No. Check: $20 + 8 = 20$ ? No. However, notice that $AB = BC = \sqrt{20}$. Let's check if $AC$ is the shortest side. $AC^2 + BC^2 = AB^2$ is not possible since $AB = BC$. Let's recalculate the distances. $AB = \sqrt{(2 - (-2))^2 + (3 - 1)^2} = \sqrt{16 + 4} = \sqrt{20}$ $BC = \sqrt{(0 - 2)^2 + (-1 - 3)^2} = \sqrt{4 + 16} = \sqrt{20}$ $AC = \sqrt{(0 - (-2))^2 + (-1 - 1)^2} = \sqrt{4 + 4} = \sqrt{8}$ So, $AC^2 + BC^2 = (\sqrt{8})^2 + (\sqrt{20})^2 = 8 + 20 = 28$ $AB^2 = (\sqrt{20})^2 = 20$ $AC^2 + AB^2 = (\sqrt{8})^2 + (\sqrt{20})^2 = 8 + 20 = 28$ $BC^2 = (\sqrt{20})^2 = 20$ $AB^2 + BC^2 = 20 + 20 = 40$ $AC^2 = 8$ So, the triangle is NOT right-angled. There must be an error in the question.

Amended Question: The points $A(-2, 1)$, $B(2, 3)$, and $C(0, -1)$ are the vertices of a triangle. Show that triangle $ABC$ is a right-angled triangle.

• Step 1: Calculate the length of side $AB$ using the distance formula: $AB = \sqrt{(2 - (-2))^2 + (3 - 1)^2}$ $AB = \sqrt{(4)^2 + (2)^2}$ $AB = \sqrt{16 + 4}$ $AB = \sqrt{20}$
• Step 2: Calculate the length of side $BC$ using the distance formula: $BC = \sqrt{(0 - 2)^2 + (-1 - 3)^2}$ $BC = \sqrt{(-2)^2 + (-4)^2}$ $BC = \sqrt{4 + 16}$ $BC = \sqrt{20}$
• Step 3: Calculate the length of side $AC$ using the distance formula: $AC = \sqrt{(0 - (-2))^2 + (-1 - 1)^2}$ $AC = \sqrt{(2)^2 + (-2)^2}$ $AC = \sqrt{4 + 4}$ $AC = \sqrt{8}$
• Step 4: Check if Pythagoras' theorem holds: $AC^2 + BC^2 = AB^2$ $(\sqrt{8})^2 + (\sqrt{20})^2 = (\sqrt{20})^2$ $8 + 20 = 28$ $AB^2 = (\sqrt{20})^2 = 20$ $AC^2 = (\sqrt{8})^2 = 8$ $BC^2 = (\sqrt{20})^2 = 20$ $AC^2 + BC^2 = 8 + 20 = 28$ $AB^2 = 20$ $AC^2 + AB^2 = 8 + 20 = 28$ $BC^2 = 20$ $AB^2 + BC^2 = 20 + 20 = 40$ $AC^2 = 8$ So, the triangle is NOT right-angled. There must be an error in the question.

Answer: The triangle is not right-angled.

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Key Equations

$$(x, y)$$

• $x$: Horizontal displacement from origin.
• $y$: Vertical displacement from origin.

Formulas for related coordinate geometry (Memorize these):

• Midpoint formula: $$(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$$
• Distance formula: $$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ (based on Pythagoras' Theorem)

Note: These formulas are generally NOT provided on the IGCSE formula sheet and should be memorized.

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Common Mistakes to Avoid

• ❌ Wrong: Swapping the order of coordinates, e.g., plotting $(4, 2)$ when the question asks for $(2, 4)$.
  • ✓ Right: Always follow the "along the hall ($x$), then up the stairs ($y$)" rule. Remember, $x$ comes before $y$ in the alphabet.
• ❌ Wrong: Miscounting units, especially when the axes have different scales (e.g., x-axis increments of 1, y-axis increments of 2).
  • ✓ Right: Carefully check the scale on both axes before reading or plotting any points.
• ❌ Wrong: Confusing points on the axes. For example, plotting $(3, 0)$ on the y-axis instead of the x-axis.
  • ✓ Right: Remember that if the $y$-coordinate is zero, the point lies on the $x$-axis. If the $x$-coordinate is zero, the point lies on the $y$-axis.
• ❌ Wrong: Forgetting the negative sign when plotting points in the 2nd or 3rd quadrant.
  • ✓ Right: Pay close attention to the signs of the $x$ and $y$ coordinates. Points in the 2nd quadrant have a negative $x$ and a positive $y$. Points in the 3rd quadrant have both negative $x$ and $y$.

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Exam Tips

• Command Words:
  • "Plot": Use a sharp HB pencil and mark the position with a small, neat 'x'.
  • "Write down": Simply state the coordinates in $(x, y)$ format.
• Accuracy Matters: Ensure your points are precisely on the grid intersections. Marks are often lost for being "half a square" off. Use a ruler to help you align points accurately.
• Types of Questions: You may be asked to plot three vertices of a square or rectangle and "State the coordinates of the fourth vertex." Always sketch the shape to visualize the missing point.
• Calculator Tip: For basic plotting, calculators are not needed. However, if you are calculating midpoints or distances between coordinates, use your calculator to double-check negative number operations (e.g., $-2 - (-5) = 3$). Be especially careful with squaring negative numbers.
• Contexts: You might see coordinates used in map-style questions where the x-axis represents Eastings and the y-axis represents Northings. Always label your axes clearly in such questions.