Gradient, Distance and Midpoint

Study Notes

The topic covers how to calculate the distance between two points and find the midpoint of a line segment using geometric principles.

• Distance Between Two Points — the length of the line segment connecting two points, calculated using the Pythagorean Theorem. Example: For points (x1, y1) and (x2, y2), the distance is $\sqrt{(x2-x1)^2 + (y2-y1)^2}$.
• Midpoint of a Line Segment — the point that divides a line segment into two equal parts. Example: For points (x1, y1) and (x2, y2), the midpoint is $\left( \frac{x1+x2}{2}, \frac{y1+y2}{2} \right)$.

Exam Tips

Key Definitions to Remember

• Distance Between Two Points
• Midpoint of a Line Segment

Common Confusions

• Forgetting to square the differences in coordinates when using the distance formula
• Mixing up the formula for midpoint with that for distance

Typical Exam Questions

• How do you calculate the distance between two points? Use the formula $\sqrt{(x2-x1)^2 + (y2-y1)^2}$.
• How do you find the midpoint of a line segment? Use the formula $\left( \frac{x1+x2}{2}, \frac{y1+y2}{2} \right)$.
• What is the distance between points (3, 4) and (7, 1)? Calculate using the distance formula to find $5$.

What Examiners Usually Test

• Correct application of the distance formula
• Accurate calculation of the midpoint