Properties and Solution of Triangle: Formulas, Rules and Notes for JEE

The solution of a triangle means finding its unknown sides and angles from the given ones, and the properties of a triangle give the formulas to do it. This page is a formula-based set of notes on the solution of triangle for JEE preparation, in the order of the topics. If your trigonometry basics are weak, first check the trigonometry formulas page, and read the next topic, heights and distances, after this one. This chapter is part of the JEE Advanced syllabus (relations between the sides and angles of a triangle, the sine rule, the cosine rule, the half-angle formula and the area of a triangle), so check the official syllabus of your exam for the topics that you need.

Notation for the Solution of Triangle

In a triangle ABC, $a$, $b$ and $c$ are the sides opposite to the angles $A$, $B$ and $C$.

Properties and Solution of Triangle: Sine Rule

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$

So $a = 2R\sin A$, $b = 2R\sin B$ and $c = 2R\sin C$. Use the sine rule when you know a side and its opposite angle, or when $R$ is in the problem.

Properties and Solution of Triangle: Cosine Rule

$$a^2 = b^2 + c^2 - 2bc\cos A, \qquad \cos A = \frac{b^2 + c^2 - a^2}{2bc}$$

with similar results for $b^2$ and $c^2$. Use it for the SAS and SSS cases.

Properties and Solution of Triangle: Projection Formulas

$$a = b\cos C + c\cos B, \qquad b = c\cos A + a\cos C, \qquad c = a\cos B + b\cos A$$

Properties and Solution of Triangle: Napier's Analogy (Tangent Rule)

$$\tan\frac{B - C}{2} = \frac{b - c}{b + c}\cot\frac{A}{2}, \quad \tan\frac{C - A}{2} = \frac{c - a}{c + a}\cot\frac{B}{2}, \quad \tan\frac{A - B}{2} = \frac{a - b}{a + b}\cot\frac{C}{2}$$

Properties and Solution of Triangle: Half-Angle Formulas

$$\sin\frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{bc}}, \qquad \cos\frac{A}{2} = \sqrt{\frac{s(s-a)}{bc}}, \qquad \tan\frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} = \frac{\Delta}{s(s-a)}$$

Properties and Solution of Triangle: Area of a Triangle

Circumradius, Inradius and Ex-radii in the Solution of Triangle

Circumradius:

$$R = \frac{a}{2\sin A} = \frac{abc}{4\Delta}$$

Inradius:

$$r = \frac{\Delta}{s} = (s - a)\tan\frac{A}{2} = 4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}$$

Ex-radii (the radius of the excircle opposite to A):

$$r_1 = \frac{\Delta}{s - a} = s\tan\frac{A}{2} = 4R\sin\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}$$

Important identities:

$$r_1 + r_2 + r_3 - r = 4R, \qquad \frac{1}{r} = \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3}$$ $$r_1r_2 + r_2r_3 + r_3r_1 = s^2, \qquad r\,r_1r_2r_3 = \Delta^2, \qquad R \geq 2r$$

How to Solve a Triangle: The Four Cases

Solved Examples on the Solution of Triangle

Example 1: All the elements of a 13-14-15 triangle

Find $\Delta$, $R$, $r$ and the ex-radii of a triangle with $a = 14$, $b = 15$, $c = 13$.

$$s = \frac{14 + 15 + 13}{2} = 21, \qquad \Delta = \sqrt{21\times 7\times 6\times 8} = \sqrt{7056} = 84$$ $$R = \frac{abc}{4\Delta} = \frac{14\times 15\times 13}{4\times 84} = 8.125, \qquad r = \frac{\Delta}{s} = \frac{84}{21} = 4$$ $$r_1 = \frac{84}{7} = 12, \qquad r_2 = \frac{84}{6} = 14, \qquad r_3 = \frac{84}{8} = 10.5$$

Check: $r_1 + r_2 + r_3 - r = 12 + 14 + 10.5 - 4 = 32.5 = 4R$.

Example 2: An angle from three sides

In a triangle with $a = 7$, $b = 8$, $c = 9$, find $\cos A$.

$$\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{64 + 81 - 49}{2\times 8\times 9} = \frac{96}{144} = \frac{2}{3}$$

Example 3: A side from two angles and a side

In a triangle, $A = 30^\circ$, $B = 45^\circ$ and $a = 10$. Find $b$.

$$b = \frac{a\sin B}{\sin A} = \frac{10\times\sin 45^\circ}{\sin 30^\circ} = \frac{10\times 0.7071}{0.5} = 10\sqrt{2} \approx 14.14$$

Example 4: A right triangle and an equilateral triangle

For a right triangle with sides 3, 4 and 5, $R = \dfrac{5}{2} = 2.5$ (half the hypotenuse) and $r = \dfrac{3 + 4 - 5}{2} = 1$. For an equilateral triangle of side $a$, $R = \dfrac{a}{\sqrt{3}}$ and $r = \dfrac{a}{2\sqrt{3}}$, so $R = 2r$, which is the case of equality in $R \geq 2r$.

Common Mistakes in the Solution of Triangle

More Notes for JEE Maths

For the full list of formulas, see eSaral's trigonometry formulas for Class 11, and the Class 12 Maths notes for JEE. For free video lectures and study material, download the eSaral app.

Handwritten Notes on the Properties and Solution of Triangle

These are eSaral's handwritten notes on the chapter, as images.