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CDS Mathematics · Formula sheet

Quadrilaterals formulas

9 formulas and 12 common traps for CDS Mathematics Quadrilaterals, grouped by subtopic.

Full notes with worked examples

Any Quadrilateral: Diagonals, Triangles and Areas

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Lengths through a diagonal

Range of a diagonal

∣a−b∣<x<a+b|a - b| < x < a + b

Rules that hold in every quadrilateral

Area on a shared diagonal

[CBD][ABD]=OCAO\dfrac{[CBD]}{[ABD]} = \dfrac{OC}{AO}

Common traps

One triangle is not enough

Each triangle gives its own range for the diagonal. The answer is where the two ranges OVERLAP; either range alone is too wide and matches a wrong option.

Heights, not squares

Triangles on a common base compare by HEIGHT, a straight ratio. Squaring the ratio belongs to similar triangles, which these are not.

Parallelograms and Midpoint Figures

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Sides, angles and area

Area

S=absin⁡θS = ab\sin\theta

Pieces cut by diagonals and midpoints

Midpoint parallelogram

[PQRS]=12[ABCD],perimeter=d1+d2[PQRS] = \tfrac12[ABCD], \quad \text{perimeter} = d_1 + d_2

Common traps

sin 150° is ½

An obtuse angle gives the same area as its supplement: sin⁡150∘=sin⁡30∘=12\sin 150^\circ = \sin 30^\circ = \tfrac12. Using the cosine, or a negative value, loses the factor pairs.

The midpoint figure's perimeter is the WHOLE sum of the diagonals

Each side of the midpoint parallelogram is half a diagonal, and there are two of each. So its perimeter is d1+d2d_1 + d_2, not half of it.

A rhombus inside a rectangle, not a square

Midpoints of a rectangle's sides give equal sides (the diagonals are equal) but not right angles, unless the rectangle is a square.

Rhombus and Kite

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Diagonals, side and area

Rhombus

d12+d22=4s2,S=12d1d2d_1^2 + d_2^2 = 4s^2, \qquad S = \tfrac12 d_1 d_2

Common traps

Half-diagonals are the legs

The right triangle uses HALF of each diagonal. Diagonals of 1010 and 2424 give a side of 1313, not 2626.

The side alone can be enough

Given the side, d12+d22=4s2d_1^2 + d_2^2 = 4s^2 is fixed for EVERY rhombus. In data-sufficiency items that means no statement is needed.

Trapeziums

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Diagonals and the four triangles

Diagonal ratio

AOOC=BOOD=ABCD\dfrac{AO}{OC} = \dfrac{BO}{OD} = \dfrac{AB}{CD}

The midline and special trapeziums

Midline

EF=AB+CD2EF = \dfrac{AB + CD}{2}

Common traps

Lengths go straight, areas go squared

The diagonals cut each other in the ratio AB:CDAB : CD. The AREAS of the triangles on the parallel sides go as its square. Using one where the other is asked is the standard wrong option.

Which sides are parallel?

Some items make AD∥BCAD \parallel BC instead of AB∥CDAB \parallel CD. The similar triangles are then AODAOD and COBCOB; relabel before using the rules.

Equal legs do not make it a trapezium

With one pair of sides parallel and the other pair equal, the figure may still be a parallelogram. Only when the equal sides are NOT parallel is it an isosceles trapezium, and so cyclic.

Cyclic Quadrilaterals

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Opposite angles and which shapes are cyclic

Opposite angles

∠A+∠C=∠B+∠D=180∘\angle A + \angle C = \angle B + \angle D = 180^\circ

Similar triangles in a cyclic quadrilateral

Across the diagonals

[APB][DPC]=(ABDC)2\dfrac{[APB]}{[DPC]} = \left(\dfrac{AB}{DC}\right)^2

Common traps

OPPOSITE angles, not adjacent ones

∠A+∠C\angle A + \angle C and ∠B+∠D\angle B + \angle D are 180∘180^\circ. ∠A+∠B\angle A + \angle B is not, unless the figure also has parallel sides.

The vertex order in a similarity statement

△EBC\triangle EBC is similar to △EDA\triangle EDA, not to △EAD\triangle EAD, because BB matches DD. Papers have written the pair in a different order; decide whether a statement means 'these triangles are similar' or 'in this correspondence' before marking it.

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