The midpoint is the point that divides a line segment into two equal parts. It lies exactly halfway between the two endpoints.

Formula

For two endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$, the midpoint $M$ is:

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

Worked Example

Find the midpoint of the segment joining $A(-2, 6)$ and $B(4, -2)$.

Solution:

$M = \left( \frac{-2 + 4}{2}, \frac{6 + (-2)}{2} \right)$ $M = \left( \frac{2}{2}, \frac{4}{2} \right)$ $M = (1, 2)$

Parallel lines are lines in the same plane that never meet, no matter how far they are extended. In coordinate geometry, parallel lines have the same slope (or gradient).

Key Property

If two lines are parallel, their slopes are equal:

$m_1 = m_2$

Worked Example

Determine if the line through $A(1, 2)$ and $B(5, 10)$ is parallel to the line through $C(3, 5)$ and $D(7, 13)$.

Solution:

Slope of line AB: $m_1 = \frac{10 - 2}{5 - 1} = \frac{8}{4} = 2$

Slope of line CD: $m_2 = \frac{13 - 5}{7 - 3} = \frac{8}{4} = 2$

Since $m_1 = m_2 = 2$, the lines are parallel.

Perpendicular lines intersect at a right angle (90°). In coordinate geometry, there is a special relationship between their slopes.

Key Property

If two lines are perpendicular, the product of their slopes equals -1:

$m_1 \times m_2 = -1$

This means the slopes are negative reciprocals of each other.

Worked Example

Show that triangle with vertices $A(1, 1)$, $B(5, 3)$, and $C(3, 7)$ is a right-angled triangle.

Solution:

Calculate slopes:

$m_{AB} = \frac{3 - 1}{5 - 1} = \frac{2}{4} = \frac{1}{2}$

$m_{BC} = \frac{7 - 3}{3 - 5} = \frac{4}{-2} = -2$

Check the product: $m_{AB} \times m_{BC} = \frac{1}{2} \times (-2) = -1$

Since the product equals -1, lines AB and BC are perpendicular. Therefore, angle B is 90° and the triangle is right-angled.

• Distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• Midpoint formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
• Parallel lines: slopes are equal ($m_1 = m_2$)
• Perpendicular lines: product of slopes equals -1 ($m_1 \times m_2 = -1$)

In Tanzania, coordinate geometry is used in surveying land for agriculture and urban planning. For example, when a farmer wants to divide a rectangular piece of land in Morogoro equally between two family members, the midpoint formula helps locate the exact center point where a dividing fence should be placed. Similarly, construction engineers use the concepts of parallel and perpendicular lines to ensure buildings are properly aligned and walls meet at right angles.