IEB · Grade 12 · Mathematics Paper 2 · Analytical geometry
In the print dialog, choose Save as PDF. IEB Grade 12 Analytical Geometry Cheat Sheet Analytical geometry is about 40 marks of Paper 2. These are the formulas and proof rules you need. Lines $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ ; $M = (\frac{x _{1} + x _{2}}{2}; \frac{y _{1} + y _{2}}{2})$ ; $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ $y - y_{1} = m (x - x_{1})$ or $y = m x + c$ . Parallel: $m_{1} = m_{2}$ . Perpendicular: $m_{1} m_{2} = - 1$ (when both gradients exist; a vertical line is perpendicular to a horizontal one). $m = tan θ$ , $0° \leq θ < 180°$ ; add 180° to a negative calculator angle. Angle between lines: difference of inclinations (exterior angle of a triangle). Circles Centre origin: $x^{2} + y^{2} = r^{2}$ . Centre $(a; b)$ : $(x - a)^{2} + (y - b)^{2} = r^{2}$ . Expanded form: complete the square for $x$ and $y$ ; the right side must be positive. Tangent at $P$ : perpendicular to radius $C P$ . Tangent length from $T$ : $T C^{2} - r^{2}$ . Line is a tangent if substituting gives $Δ = 0$ . Polygon proofs (minimum conditions) Parallelogram: diagonals share a midpoint. Rhombus: parallelogram with adjacent sides equal (or perpendicular diagonals). Rectangle: parallelogram with a right angle (or equal diagonals). Square: rhombus with a right angle. Collinear: equal gradients through a shared point. End with a sentence naming the property used. Euclidean links Angle in a semicircle: a chord subtending 90° is a diameter. Line from centre to midpoint of a chord is perpendicular to it. Know by heart Minimum conditions for each quadrilateral Completing the square for circles On the formula sheet Distance, midpoint, gradient, line and circle equations, $m = tan θ$ Common traps Centre of $(x - 3)^{2} + (y + 2)^{2} = 25$ written as $(- 3; 2)$ . Radius given as $r^{2}$ . Using the radius gradient as the tangent gradient. Negative angle of inclination.
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IEB Grade 12 Analytical Geometry Cheat Sheet

Analytical geometry is about 40 marks of Paper 2. These are the formulas and proof rules you need.

Lines

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ ;
$M = (\frac{x _{1} + x _{2}}{2}; \frac{y _{1} + y _{2}}{2})$ ;
$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
$y - y_{1} = m (x - x_{1})$ or $y = m x + c$ .
Parallel: $m_{1} = m_{2}$ . Perpendicular: $m_{1} m_{2} = - 1$ (when both gradients exist; a vertical line is perpendicular to a horizontal one).
$m = tan θ$ , $0° \leq θ < 180°$ ; add 180° to a negative calculator angle.
Angle between lines: difference of inclinations (exterior angle of a triangle).

Circles

Centre origin: $x^{2} + y^{2} = r^{2}$ . Centre $(a; b)$ : $(x - a)^{2} + (y - b)^{2} = r^{2}$ .
Expanded form: complete the square for $x$ and $y$ ; the right side must be positive.
Tangent at $P$ : perpendicular to radius $C P$ .
Tangent length from $T$ : $T C^{2} - r^{2}$ .
Line is a tangent if substituting gives $Δ = 0$ .

Polygon proofs (minimum conditions)

Parallelogram: diagonals share a midpoint.
Rhombus: parallelogram with adjacent sides equal (or perpendicular diagonals).
Rectangle: parallelogram with a right angle (or equal diagonals).
Square: rhombus with a right angle.
Collinear: equal gradients through a shared point.
End with a sentence naming the property used.

Euclidean links

Angle in a semicircle: a chord subtending 90° is a diameter.
Line from centre to midpoint of a chord is perpendicular to it.

Know by heart

Minimum conditions for each quadrilateral
Completing the square for circles

On the formula sheet

Distance, midpoint, gradient, line and circle equations, $m = tan θ$

Common traps

Centre of $(x - 3)^{2} + (y + 2)^{2} = 25$ written as $(- 3; 2)$ .
Radius given as $r^{2}$ .
Using the radius gradient as the tangent gradient.
Negative angle of inclination.