Gradient, Midpoint & Length

Three formulas underpin all of coordinate geometry. For $A(x_1, y_1)$ and $B(x_2, y_2)$:

• Gradient: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$
• Midpoint: $\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)$
• Length: $AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

None are given in the exam (memorise list); all three must run without thought.

The consistency rule

The gradient’s one trap is mixing subtraction orders: $(y_2 - y_1)$ over $(x_1 - x_2)$ silently negates the answer. Fix it structurally, label the points on the question paper, write the formula with the same point first in both numerator and denominator, then substitute:

$A(2, -3)$, $B(6, 5)$: $m = \dfrac{5 - (-3)}{6 - 2} = \dfrac{8}{4} =$ $2$

Negative coordinates are where this pays: the double negative in $(5 - (-3))$ is visible and checkable on paper, invisible and fragile in the head, especially on the non-calculator paper.

Lengths stay exact

$AB$ for $A(1, 2)$, $B(4, 8)$: $\sqrt{9 + 36} = \sqrt{45} =$ $3\sqrt{5}$

On Paper 1, simplify the surd and stop. $6.71$ is an approximation nobody asked for. Length answers feed later parts (areas, perimeters, $\text{radius}^2$ in circle equations), and only exact forms survive the journey without rounding damage. Often $\text{length}^2$ is all you need (comparing distances, Pythagoras checks), skip the square root entirely and say so.

What gets built from the primitives

• Gradient → equation of a line, parallel/perpendicular tests
• Midpoint → perpendicular bisectors, centres from diameters
• Length → distances, radii, “show the triangle is isosceles” (two equal $\text{length}^2$ values, plus the stated conclusion)

“Show $ABC$ is right-angled” can run two ways: gradients ($m_1 m_2 = -1$, usually faster) or lengths (Pythagoras on the three sides). Pick one, state the test, conclude in words.

Common mistakes

• Subtraction order mixed between numerator and denominator
• Midpoint computed with differences instead of sums
• Surds decimalised mid-question
• Length used where $\text{length}^2$ suffices (extra surd work, extra risk)
• Conclusions (“hence isosceles”) left unstated after correct computation

Full topic context: Straight-Line Graphs notes.