JEE Mains Maths · Properties of Triangle

Solution of Triangles

Relating the sides and angles of a triangle through the sine and cosine rules, and its area to the radii of its incircle and circumcircle.

All Properties of Triangle notes JEE Mains Maths strategy

Why this matters

Eleven PYQs, six of them multiple choice, from 2021 to 2024. Five solve a triangle with the sine or cosine rule; three turn a condition on the angles into one on the sides; three use the area, the inradius and the circumradius together. Three ideas cover the page.

Concept 1 of 3: The sine and cosine rules

The sine rule links each side with the angle opposite it and with the circumradius; the cosine rule finds an angle from three sides, or the third side from two sides and the angle between them. Given a side and an angle that is not between the known sides, the cosine rule gives a quadratic, and both roots may need checking.

Definition

Sine rule: $\frac a{\sin A}=\frac b{\sin B}=\frac c{\sin C}=2R$ .
Cosine rule: $a^2=b^2+c^2-2bc\cos A$ .
Angles add to $\pi$ ; the larger side faces the larger angle.

Cosine rule

a^2=b^2+c^2-2bc\cos A

Worked example

In a triangle,

b=5

c=8

and

A=60^\circ

. Find

a

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 6 April 2024 · Q176Moderate

Example 1 · Properties of Triangle · Solution of Triangles

In a triangle

ABC,BC = 7,AC = 8,AB =\alpha\in N

and

\cos A =\frac{2}{3}

. If

49\cos(3C) + 42 =\frac{m}{n}

, where

gcd(m,n) = 1

, then

m + n

is equal to

Check both roots

The cosine rule for an unknown side next to a known angle is a quadratic. A negative root is impossible, but two positive roots can both be triangles; the question's other data decide.

Drill 4 more on the sine and cosine rules

Concept 2 of 3: Conditions on the angles

A condition like

\cos A+2\cos B+\cos C=2

\frac{\sin A}{\sin B}=\dots

becomes a condition on the sides once each sine is replaced by the side over

2R

, or each cosine by its cosine-rule expression. The half-angle formulas turn products of

\sin\frac A2

into the inradius.

Definition

$\sin A=\frac a{2R}$ , and similarly for $B,C$ .
$\cos A+\cos B+\cos C=1+\frac rR$ .
$\sin\frac A2\sin\frac B2\sin\frac C2=\frac r{4R}$ .
Projection rule: $a=b\cos C+c\cos B$ .

Sines to sides

\sin A=\frac{a}{2R}

Worked example

In a triangle,

a\cos A=b\cos B

with

a\neq b

. What kind of triangle is it?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 12 Apr 2023 · Q80Moderate

Example 2 · Properties of Triangle · Solution of Triangles

In a triangle

ABC

, if

\cos A + 2\cos B + \cos C = 2

and the lengths of the sides opposite to the angles

A

and

C

are 3 and 7 respectively, then

\cos A - \cos C

is equal to

sin 2A = sin 2B has two cases

\sin2A=\sin2B

gives

A=B

A+B=\frac\pi2

. Discard one only when the data rule it out.

Drill 2 more on conditions on the angles

Concept 3 of 3: Area, inradius and circumradius

The area

\Delta

connects everything:

\Delta=rs

with the semi-perimeter

s

, and

\Delta=\frac{abc}{4R}

. Find the sides (or their ratio), the area by Heron's formula or

\frac12bc\sin A

, and then

r

and

R

follow.

Definition

$\Delta=\frac12bc\sin A=\sqrt{s(s-a)(s-b)(s-c)}$ .
$r=\frac\Delta s$ , $R=\frac{abc}{4\Delta}$ .
In a right triangle, $R$ is half the hypotenuse and $r=s-c$ .

The two radii

r=\frac{\Delta}{s},\qquad R=\frac{abc}{4\Delta}

Worked example

Find

r

and

R

for the triangle with sides 13, 14, 15.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 25 June 2022 · Q63Moderate

Example 3 · Properties of Triangle · Solution of Triangles

Let

a,b

and

c

be the length of sides of a triangle

ABC

such that

\frac{a + b}{7}=\frac{b + c}{8}=\frac{c + a}{9}

. If

r

and

R

are the radius of incircle and radius of circumcircle of the triangle

ABC

, respectively, then the value of

\frac{R}{r}

is equal to

Semi-perimeter, not perimeter

r=\frac\Delta s

uses half the perimeter. Dividing by the full perimeter halves the inradius.

Drill 2 more on area, inradius and circumradius

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 (3)

The sine and cosine rules
Cosine rule
$a^2=b^2+c^2-2bc\cos A$
Conditions on the angles
Sines to sides
$\sin A=\frac{a}{2R}$
Area, inradius and circumradius
The two radii
$r=\frac{\Delta}{s},\qquad R=\frac{abc}{4\Delta}$

Watch out for (3)

Check both roots
→ The sine and cosine rules
sin 2A = sin 2B has two cases
→ Conditions on the angles
Semi-perimeter, not perimeter
→ Area, inradius and circumradius

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