Quadrilaterals — CDS Elementary Mathematics

Formulas (2)

• Lengths through a diagonal · Range of a diagonal
  $$|a - b| < x < a + b$$
• Rules that hold in every quadrilateral · Area on a shared diagonal
  $$\dfrac{[CBD]}{[ABD]} = \dfrac{OC}{AO}$$

Watch out for (2)

• One triangle is not enough
  → Lengths through a diagonal
• Heights, not squares
  → Rules that hold in every quadrilateral

Formulas (2)

• Sides, angles and area · Area
  $$S = ab\sin\theta$$
• Pieces cut by diagonals and midpoints · Midpoint parallelogram
  $$[PQRS] = \tfrac12[ABCD], \quad \text{perimeter} = d_1 + d_2$$

Watch out for (3)

• sin 150° is ½
  → Sides, angles and area
• The midpoint figure's perimeter is the WHOLE sum of the diagonals
  → Pieces cut by diagonals and midpoints
• A rhombus inside a rectangle, not a square
  → Pieces cut by diagonals and midpoints

Formulas (1)

• Diagonals, side and area · Rhombus
  $$d_1^2 + d_2^2 = 4s^2, \qquad S = \tfrac12 d_1 d_2$$

Watch out for (2)

• Half-diagonals are the legs
  → Diagonals, side and area
• The side alone can be enough
  → Diagonals, side and area

Formulas (2)

• Diagonals and the four triangles · Diagonal ratio
  $$\dfrac{AO}{OC} = \dfrac{BO}{OD} = \dfrac{AB}{CD}$$
• The midline and special trapeziums · Midline
  $$EF = \dfrac{AB + CD}{2}$$

Watch out for (3)

• Lengths go straight, areas go squared
  → Diagonals and the four triangles
• Which sides are parallel?
  → Diagonals and the four triangles
• Equal legs do not make it a trapezium
  → The midline and special trapeziums

Formulas (2)

• Opposite angles and which shapes are cyclic · Opposite angles
  $$\angle A + \angle C = \angle B + \angle D = 180^\circ$$
• Similar triangles in a cyclic quadrilateral · Across the diagonals
  $$\dfrac{[APB]}{[DPC]} = \left(\dfrac{AB}{DC}\right)^2$$

Watch out for (2)

• OPPOSITE angles, not adjacent ones
  → Opposite angles and which shapes are cyclic
• The vertex order in a similarity statement
  → Similar triangles in a cyclic quadrilateral