7.3 Line Geometry Problems

2026 📋 Syllabus Objectives

By the end of these notes, you will be able to:

  • Find the midpoint of a line segment joining two points
  • Find the length of a line segment joining two points
  • Solve problems that use the midpoint and length formulas
  • Find and use the equation of a perpendicular bisector

📐 The Essential Formulas (Your Toolkit)

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Before diving into problems, you need to have four key formulas memorised. Everything in this topic comes back to these.

Let's say you have two points: P(x₁, y₁) and Q(x₂, y₂).

Formula 1 — Length of a Line Segment

PQ=(x2−x1)2+(y2−y1)2PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

What this means in plain English: To find the distance between two points, subtract the x-coordinates and square the result, subtract the y-coordinates and square that result, add both squared values together, then take the square root.

💡 Think of it like a right-angled triangle. The line PQ is the hypotenuse (longest side), and the horizontal and vertical gaps are the other two sides. You're just using Pythagoras' theorem!

Formula 2 — Midpoint of a Line Segment

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

What this means in plain English: The midpoint (the exact middle of the line) is found by taking the average of the two x-coordinates, and the average of the two y-coordinates.

💡 Average just means: add them together and divide by 2.

Formula 3 — Gradient of a Line

gradient=y2−y1x2−x1\text{gradient} = \frac{y_2 - y_1}{x_2 - x_1}

What this means in plain English: Gradient tells you how steep a line is. Divide the vertical change (up or down) by the horizontal change (left or right).

Formula 4 — Gradient of a Perpendicular Line

If one line has gradient m, then any line perpendicular (at a right angle, 90°) to it has gradient:

m⊥=−1mm_{\perp} = -\frac{1}{m}

This can also be written as:

m1×m2=−1m_1 \times m_2 = -1

What this means in plain English: Flip the fraction and change the sign (positive becomes negative, negative becomes positive). The two gradients multiplied together always equal −1.

💡 Example: If a line has gradient 3, a perpendicular line has gradient −⅓. Check: 3 × (−⅓) = −1 ✓

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