IEB Grade 11 Analytical Geometry Cheat Sheet

Analytical geometry is about 30 marks of Grade 11 Paper 2. These are the formulas and proof rules.

Formulas

• $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$;
  $M=\left(\frac{x_1+x_2}{2};\frac{y_1+y_2}{2}\right)$;
  $m=\frac{y_2-y_1}{x_2-x_1}$

• $y-y_1=m(x-x_1)$; parallel $m_1=m_2$; perpendicular $m_1{m}_2=-1$ (when both gradients exist; a vertical line is perpendicular to a horizontal one).

• $m=\tan \theta$; add 180° to a negative calculator angle.

• Angle between lines: larger inclination $-$ smaller inclination.

Special lines in a triangle

• Median: vertex to the midpoint of the opposite side.

• Altitude: from a vertex, perpendicular to the opposite side.

• Perpendicular bisector: through a midpoint, perpendicular to that side.

Proofs

• Collinear: equal gradients through a shared point.

• Right angle: gradients multiply to $-1$. Isosceles: two equal sides.

• Parallelogram: diagonals share a midpoint. Rhombus: plus adjacent sides equal. Rectangle: plus a right angle. Square: rhombus plus a right angle. Kite: two pairs of equal adjacent sides.

• Always end with a sentence naming the property.

Know by heart

• The difference between median, altitude and perpendicular bisector

• Minimum conditions for each quadrilateral

On the formula sheet

• Distance, midpoint, gradient and line equations, $m=\tan \theta$

Common traps

• Using the side's gradient instead of the perpendicular gradient for an altitude.

• Negative angles of inclination.

• Leaving out the conclusion.