JEE Mains Maths · Straight Lines

Area of Triangles and Quadrilaterals

The area of a triangle from its vertices, collinearity, ratios of areas inside a triangle, and parallelograms, rhombi, squares and isosceles triangles built from given lines and points.

All Straight Lines notes JEE Mains Maths strategy

Why this matters

Nineteen PYQs, twelve of them multiple choice, and one from 2026. Eleven compute an area from coordinates or compare areas inside a triangle. Eight build a parallelogram, a rhombus, a square or an isosceles triangle from given lines and points and ask for a vertex or a length. Two ideas cover the page.

Concept 1 of 2: Area from coordinates

The area of a triangle is half the modulus of

x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)

. It is zero exactly when the three points are collinear. Because of the modulus, a given area gives two equations. Ratios of areas often need no coordinates: triangles with the same height have areas in the ratio of their bases, and the triangle joining the midpoints of the sides has a quarter of the area.

Definition

Area $=\frac12|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$ .
Collinear exactly when the bracket is 0.
Same height: areas in the ratio of the bases.
Triangle of midpoints: $\frac14$ of the area.

Area of a triangle

\Delta=\frac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|

Worked example

Find the area of the triangle with vertices

(1,2)

(4,-2)

(6,5)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 2 Apr 2025 · Q149Moderate

Example 1 · Straight Lines · Area of Triangles and Quadrilaterals

Let

A(4, - 2),B(1,1)

and

C(9, - 3)

be the vertices of a triangle

ABC

. Then the maximum area of the parallelogram AFDE , formed with vertices

D,E

and

F

on the sides

BC,CA

and

AB

of the triangle ABC respectively, is

Two signs from one modulus

An area of 4 means the bracket is

8

-8

. Dropping one sign loses half the answers, which matters when the question asks for the sum of all values.

Drill 10 more on area from coordinates

Concept 2 of 2: Parallelograms, rhombi and isosceles triangles

The diagonals of a parallelogram bisect each other, so

A+C=B+D

. A rhombus adds that its diagonals are perpendicular, and a square that they are also equal. In an isosceles triangle, the altitude from the apex is also the median, so its foot is the midpoint of the base. These facts turn a named shape into equations for the unknown vertices.

Definition

Parallelogram $ABCD$ : $A+C=B+D$ , the common midpoint of the diagonals.
Rhombus: also $AC\perp BD$ . Square: also $AC=BD$ .
Square of side $a$ : diagonal $a\sqrt2$ , at $45^\circ$ to the sides.
Isosceles with $AB=AC$ : the foot of the altitude from $A$ is the midpoint of $BC$ .

Parallelogram

A+C=B+D

Worked example

Three vertices of parallelogram

ABCD

are

A(2,1)

B(5,3)

C(7,8)

. Find

D

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 23 Jan 2026 Shift 2 · Q54Moderate

Example 2 · Straight Lines · Area of Triangles and Quadrilaterals

Let

A(1,2)

and

C( - 3, - 6)

be two diagonally opposite vertices of a rhombus, whose sides AD an BC are parallel to the line

7x - y = 14

. If

B(\alpha,\beta)

and

D(\gamma,\delta)

are the other two vertices, then

|\alpha + \beta + \gamma + \delta|

is equal to :

Which vertices are opposite

A+C=B+D

holds when

A

and

C

are opposite. If the vertices are named

ABCD

in order, they are. If only three points are given, the fourth vertex has three possible positions.

Drill 7 more on parallelograms, rhombi and isosceles triangles

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)

Area from coordinates
Area of a triangle
$\Delta=\frac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|$
Parallelograms, rhombi and isosceles triangles
Parallelogram
$A+C=B+D$

Watch out for (2)

Two signs from one modulus
→ Area from coordinates
Which vertices are opposite
→ Parallelograms, rhombi and isosceles triangles

Test yourself on Straight Lines

20 past JEE Mains questions from this chapter, timed at 48 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 Straight Lines chapter test Drill all 19 questions