Analytical Geometry Formula Sheet (Grade 12 CAPS)

Distance, midpoint, gradient, straight line, inclination and circle formulas for Grade 12 CAPS analytical geometry, with NSC Paper 2 usage notes.

Analytical geometry rewards a systematic approach: plot the points, decide which formula the question needs, and justify each conclusion with the property you used.

Formulas

Formula	Expression	Note
Distance between points	`d = √[(x₂ − x₁)² + (y₂ − y₁)²]`	Used to prove equal sides or a radius.
Midpoint	`M = ((x₁ + x₂)/2 ; (y₁ + y₂)/2)`	Diagonals of a parallelogram share a midpoint.
Gradient	`m = (y₂ − y₁)/(x₂ − x₁)`	Parallel lines have equal gradients; perpendicular gradients multiply to −1.
Equation of a line	`y − y₁ = m(x − x₁)`	Fastest form when a point and gradient are known.
Angle of inclination	`m = tanθ`	Add 180° when the gradient is negative to get 90° < θ < 180°.
Circle centred at origin	`x² + y² = r²`	Radius is √r², never r² itself.
Circle centred at (a; b)	`(x − a)² + (y − b)² = r²`	Complete the square to convert from general form.
Tangent to a circle	`m_radius × m_tangent = −1`	The tangent is perpendicular to the radius at the point of contact.

How to use them

Always sketch the points before calculating — it prevents sign errors.
Quote the property (equal gradients, equal sides) as your reason in proof questions.
For tangents, find the radius gradient first.
Worked example: the line through (2; 1) perpendicular to y = 2x + 5 has gradient −½, so y − 1 = −½(x − 2), which gives y = −½x + 2.

Frequently asked questions

Are all Analytical Geometry formulas given in the NSC exam?

Some are printed on the official information sheet, but not all. Treat every formula on this page as examinable and learn it with a worked example so you recognise when to use it.

Which NSC paper tests Analytical Geometry?

Analytical Geometry is examined in Paper 2 of the NSC Mathematics examination.

How do I practise these formulas?

Snap a analytical geometry question into the Snap&Learn AI solver — it shows the substitution and every method mark the way the NSC memo does.

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