CDS Mathematics · Quadrilaterals

Parallelograms and Midpoint Figures

A parallelogram's opposite sides are equal and parallel and its diagonals bisect each other; joining the midpoints of any quadrilateral's sides makes one.

All Quadrilaterals notes CDS Mathematics strategy

Why this matters

Fourteen PYQs, two of them HARD. Six use the sides and angles (area = ab sin θ, adjacent angles supplementary); the other eight cut the figure with its diagonals or with lines through midpoints and compare the pieces.

Concept 1 of 2: Sides, angles and area

Push a rectangle sideways and it becomes a parallelogram with the same base; the height shrinks to

b\sin\theta

. So the area is

ab\sin\theta

, and adjacent angles still add to

180^\circ

Definition

Opposite sides equal and parallel; opposite angles equal; adjacent angles supplementary.
Area $= ab\sin\theta$ for adjacent sides $a, b$ at angle $\theta$ .
The bisectors of two adjacent angles meet at $90^\circ$ .
Four equal sides: a rhombus. Equal diagonals: a rectangle.
Integer sides with a given area: list the factor pairs of $ab$ .

Area

S = ab\sin\theta

Worked example

A parallelogram has integer sides, an angle of

30^\circ

and area

11

^2

. Find its possible perimeters.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (II) 2026 — Elementary Mathematics · Q78Hard

Example 1 · Quadrilaterals · Parallelograms

ABCD is a parallelogram with

\angle ABC = 150^\circ

and the sides have integer values. If the area of the parallelogram is 17.5

\text{cm}^2

, then consider the following statements : I. It is possible to have a perimeter equal to 24 cm. II. It is possible to have a perimeter equal to 72 cm. Which of the statements given above is/are correct ?

sin 150° is ½

An obtuse angle gives the same area as its supplement:

\sin 150^\circ = \sin 30^\circ = \tfrac12

. Using the cosine, or a negative value, loses the factor pairs.

Drill 5 more on sides, angles and area

Concept 2 of 2: Pieces cut by diagonals and midpoints

The diagonals cut a parallelogram into four triangles of equal area. A line from a vertex to the midpoint of a side makes similar triangles in the ratio

1 : 2

. Joining midpoints of sides halves the area.

Definition

The diagonals bisect each other and cut the figure into four triangles of equal area.
A line from $A$ to the midpoint $E$ of $BC$ cuts diagonal $BD$ at $F$ with $BF = \dfrac{BD}{3}$ (triangles $BFE$ and $DFA$ , ratio $1 : 2$ ).
Joining the midpoints of the sides of ANY quadrilateral gives a parallelogram (Varignon) of half the area, with sides parallel to the diagonals and half as long.
Midpoints of a rectangle's sides: a rhombus. Of a rhombus's sides: a rectangle.
In a triangle, the midpoints and one vertex form a parallelogram of half the triangle's area.

Midpoint parallelogram

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

Worked example

E

is the midpoint of

BC

in parallelogram

ABCD

, and

AE

meets

BD

F

. If

BD = 9

cm, find

FD

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (I) 2026 — Elementary Mathematics · Q54Moderate

Example 2 · Quadrilaterals · Parallelograms

Let ABCD be a parallelogram and E be the midpoint of BC. The diagonal BD and line segment AE intersects at F. If

BF = 2.4

cm, then what is BD equal to ?

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

d_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.

Drill 7 more on pieces cut by diagonals and midpoints

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)

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

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 14 questions