In the realm of IGCSE maths, understanding the intricacies of coordinate geometry through practical examples is paramount. This section offers a deep dive into solving for perpendicular lines and their equations, emphasizing a procedural approach to enhance problem-solving acumen.

More on Perpendicular Lines

Perpendicular lines intersect at a right angle $90 \text{ degrees }$. The relationship between their gradients is particularly important: the product of the gradients of two perpendicular lines is always $-1$. This fundamental property is crucial in solving problems related to perpendicular lines in coordinate geometry.

Example 1:

Calculating the Gradient of a Line Perpendicular to $2y = 3x + 1$.

Solution:

• Given Line Equation

• Convert to Slope-Intercept Form

• Gradient of Given Line ((m))

• Gradient of Perpendicular Line ((m'))

Thus, the gradient of the line perpendicular to $2y = 3x + 1$ is $-\frac{2}{3}$.

Example 2:

Finding the Equation of the Perpendicular Bisector of the Line Joining Points $(-3, 8)$ and $(9, -2)$.

Solution:

• Calculate Midpoint $(M)$

• Gradient of Line Segment $(m)$

• Gradient of Perpendicular Bisector $(m')$

• Equation of Perpendicular Bisector

Using the point-slope form $y - y_1 = m'(x - x_1)$ with $M = (3, 3)$ and $m' = 1.2$:

Simplifying to the slope-intercept form $y = mx + c$:

Hence, the equation of the perpendicular bisector of the line segment joining the points $(-3, 8)$ and $(9, -2)$ is $y = 1.2x - 0.6$.