Solving Problems on Coordinate Axes

Coordinate Geometry

Coordinate geometry is all about using a grid system to describe where things are and to solve problems involving distance, midpoints, and area. It’s a key skill used in everything from designing video game levels to helping planes navigate.

Key Coordinate Geometry Formulas

1. Length of a Line Segment

To find the distance between two points, we use Pythagoras’ theorem on the right-angled triangle formed by the change in x and the change in y.

$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$

2. Midpoint of a Line Segment

The midpoint is the exact middle of a line segment. You find it by averaging the x-coordinates and averaging the y-coordinates.

$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

3. Area of a Shape

To find the area of a polygon, you can use standard formulas (like $\frac{1}{2} \times \text{base} \times \text{height}$) if a side is horizontal or vertical, or use the “box method” for tilted shapes.

Worked Examples

Example 1: Finding the Length

Find the length of the line segment joining A(1, 2) and B(4, 6).

Find change in x: $4 – 1 = 3$.
Find change in y: $6 – 2 = 4$.
Apply Pythagoras: Length$^2 = 3^2 + 4^2 = 9 + 16 = 25$.
Square root: Length = $\sqrt{25} = 5$.

Answer: 5 units.

Example 2: Finding the Midpoint

Find the midpoint of the line segment joining P(2, 7) and Q(8, 1).

Average the x-coordinates: $\frac{2 + 8}{2} = \frac{10}{2} = 5$.
Average the y-coordinates: $\frac{7 + 1}{2} = \frac{8}{2} = 4$.

Answer: (5, 4).

Tutor Insights

🤔 Common Misunderstandings

Mixing up x and y coordinates. Always remember the format is $(x, y)$.
Sign errors with negative coordinates, especially when finding differences for length.
Confusing length and midpoint formulas. Length uses Pythagoras (subtract and square), while midpoint uses averages (add and divide).

📝 Common Exam Mistakes

Forgetting to square root at the end of a length calculation.
Incorrectly applying Pythagoras by adding the squares when you should subtract (when finding a shorter side).
Calculation errors with negative numbers or fractions.
Not drawing a diagram, which can make it harder to spot the correct method.

Practice Questions

Find the length of the line segment joining P(2, 3) and Q(5, 7).
Find the midpoint of the line segment joining A(1, 8) and B(7, 2).
Calculate the length of the line segment connecting C(-3, 4) and D(5, -2).
A triangle has vertices at G(1, 2), H(7, 2), and I(4, 8). Find its area.

Show Answers

Working: Length = $\sqrt{(5-2)^2 + (7-3)^2} = \sqrt{3^2 + 4^2} = \sqrt{25}$.
Answer: 5 units.
Working: Midpoint = $(\frac{1+7}{2}, \frac{8+2}{2}) = (\frac{8}{2}, \frac{10}{2})$.
Answer: (4, 5).
Working: Length = $\sqrt{(5-(-3))^2 + (-2-4)^2} = \sqrt{8^2 + (-6)^2} = \sqrt{64+36} = \sqrt{100}$.
Answer: 10 units.
Working: Base GH is horizontal, length = $7-1=6$. Height is vertical distance from I to the base, $8-2=6$. Area = $\frac{1}{2} \times 6 \times 6 = 18$.
Answer: 18 square units.