CDS Mathematics · Quadrilaterals

Trapeziums

In a trapezium the diagonals cut each other in the ratio of the parallel sides, and the line joining the midpoints of the legs is their average.

All Quadrilaterals notes CDS Mathematics strategy

Why this matters

Eight PYQs, two of them HARD. Five use one pair of similar triangles — the two on the parallel sides, cut off by the diagonals — for ratios of lengths and, squared, of areas. The rest use the midline or an isosceles or right trapezium.

Concept 1 of 2: Diagonals and the four triangles

The parallel sides make alternate angles equal, so the triangles on them,

AOB

and

COD

, are similar in the ratio

AB : CD

. The two side triangles have equal areas, because each is a big triangle on a parallel side minus the same piece.

Definition

$AB \parallel CD$ : $\dfrac{AO}{OC} = \dfrac{BO}{OD} = \dfrac{AB}{CD}$ .
$[AOB] : [COD] = AB^2 : CD^2$ .
The side triangles are equal: $[AOD] = [BOC]$ , and $[AOD]^2 = [AOB]\cdot[COD]$ .
A line parallel to the parallel sides cuts the legs in the same ratio.

Diagonal ratio

\dfrac{AO}{OC} = \dfrac{BO}{OD} = \dfrac{AB}{CD}

Worked example

In trapezium

ABCD

AB \parallel CD

and

3AB = 5CD

. Find

[AOB] : [COD]

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — Elementary Mathematics · Q89Moderate

Example 1 · Quadrilaterals · Trapeziums

ABCD

is a trapezium in which

AB

is parallel to

DC

and

2AB = 3DC

. The diagonals

AC

and

BD

intersect at

O

. What is the ratio of the area of

\triangle AOB

to that of

\triangle DOC

Lengths go straight, areas go squared

The diagonals cut each other in the ratio

AB : 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 \parallel BC

instead of

AB \parallel CD

. The similar triangles are then

AOD

and

COB

; relabel before using the rules.

Drill 4 more on diagonals and the four triangles

Concept 2 of 2: The midline and special trapeziums

The segment joining the midpoints of the legs runs halfway up the trapezium, so its length is halfway between the two parallel sides: their average.

Definition

Midline $= \dfrac{a + b}{2}$ , where $a, b$ are the parallel sides. Area $=$ midline $\times$ height.
Isosceles trapezium (equal legs, not a parallelogram): base angles equal, and it is always cyclic.
A trapezium with an incircle: the height is the circle's diameter, and each leg equals the sum of the two tangents from its ends.

Midline

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

Worked example

A trapezium's midline is

13

cm and its parallel sides differ by

6

cm. Find the parallel sides.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (I) 2022 — Elementary Mathematics · Q73Moderate

Example 2 · Quadrilaterals · Trapeziums

ABCD

is a trapezium in which

AB

is parallel to

DC

. Let

E

and

F

be the midpoints on

AD

and

BC

respectively. If

EF = 10

cm and

AB - DC = 4

cm, then what is the value of

AB \times DC

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.

Drill 2 more on the midline and special trapeziums

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

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

Test yourself on Quadrilaterals

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Quadrilaterals chapter test Drill all 8 questions