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Properties AND Solution OF Triangles formulas

Master Properties AND Solution OF Triangles through 20 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Properties AND Solution OF Triangles, every formula

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Sine rule
$\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C} = 2 R$
Q1MCQSine rule
In any triangle, $\dfrac{a}{\sin A}$ equals:
- A
  $2R$
- B
  $R$
- C
  $\dfrac{R}{2}$
- D
  $4R$
Cosine rule
$cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}$
Q1MCQCosine rule
$\cos A$ equals:
- A
  $\dfrac{b^{2}+c^{2}-a^{2}}{2bc}$
- B
  $\dfrac{a^{2}+c^{2}-b^{2}}{2ac}$
- C
  $\dfrac{a^{2}+b^{2}-c^{2}}{2ab}$
- D
  $\dfrac{b^{2}+c^{2}}{2bc}$
Projection formula
$a = b cos C + c cos B$
Q1MCQProjection
The projection formula gives $a$ as:
- A
  $b\cos C+c\cos B$
- B
  $b\cos B+c\cos C$
- C
  $b+c$
- D
  $bc\cos A$
Area (two sides + angle)
$Δ = \frac{1}{2} ab sin C$
Area (Heron)
$Δ = s (s - a) (s - b) (s - c)$
Semi-perimeter
$s = \frac{a + b + c}{2}$
Circumradius
$R = \frac{ab c}{4Δ}$
Q1MCQCircumradius
The circumradius $R$ equals:
- A
  $\dfrac{abc}{4\Delta}$
- B
  $\dfrac{\Delta}{s}$
- C
  $\dfrac{4\Delta}{abc}$
- D
  $\dfrac{abc}{\Delta}$
Inradius
$r = \frac{Δ}{s}$
Q1MCQInradius
The inradius $r$ equals:
- A
  $\dfrac{\Delta}{s}$
- B
  $\dfrac{s}{\Delta}$
- C
  $\dfrac{abc}{4\Delta}$
- D
  $\Delta s$
Tangent half-angle
$tan \frac{A}{2} = \frac{( s - b ) ( s - c )}{s ( s - a )}$
Q1MCQtan A/2
$\tan\dfrac{A}{2}$ equals:
- A
  $\sqrt{\dfrac{(s-b)(s-c)}{s(s-a)}}$
- B
  $\sqrt{\dfrac{s(s-a)}{bc}}$
- C
  $\sqrt{\dfrac{(s-b)(s-c)}{bc}}$
- D
  $\dfrac{s-a}{s}$
Sine half-angle
$sin \frac{A}{2} = \frac{( s - b ) ( s - c )}{b c}$
Q1MCQsin A/2
$\sin\dfrac{A}{2}$ equals:
- A
  $\sqrt{\dfrac{(s-b)(s-c)}{bc}}$
- B
  $\sqrt{\dfrac{s(s-a)}{bc}}$
- C
  $\sqrt{\dfrac{(s-b)(s-c)}{s(s-a)}}$
- D
  $\dfrac{s-a}{bc}$
Cosine half-angle
$cos \frac{A}{2} = \frac{s ( s - a )}{b c}$
Q1MCQcos A/2
$\cos\dfrac{A}{2}$ equals:
- A
  $\sqrt{\dfrac{s(s-a)}{bc}}$
- B
  $\sqrt{\dfrac{(s-b)(s-c)}{bc}}$
- C
  $\dfrac{s}{bc}$
- D
  $\sqrt{s(s-a)}$
Napier's analogy
$tan \frac{B - C}{2} = \frac{b - c}{b + c} cot \frac{A}{2}$
Q1MCQNapier
Napier's analogy relates $\tan\dfrac{B-C}{2}$ to:
- A
  $\dfrac{b-c}{b+c}\cot\dfrac{A}{2}$
- B
  $\dfrac{b+c}{b-c}\cot\dfrac{A}{2}$
- C
  $\dfrac{b-c}{b+c}\tan\dfrac{A}{2}$
- D
  $\cot\dfrac{A}{2}$
Area (R form)
$Δ = \frac{ab c}{4 R}$
Q1MCQArea R form
The area in terms of $R$ is:
- A
  $\dfrac{abc}{4R}$
- B
  $\dfrac{4R}{abc}$
- C
  $abcR$
- D
  $\dfrac{abc}{R}$
m-n theorem
$(m + n) cot θ = m cot α - n cot β$
Q1MCQEscribed radius
The escribed radius $r_1$ (opposite A) is:
- A
  $\dfrac{\Delta}{s-a}$
- B
  $\dfrac{\Delta}{s}$
- C
  $\dfrac{s-a}{\Delta}$
- D
  $\dfrac{\Delta}{s+a}$
Escribed radius r₁
$r_{1} = \frac{Δ}{s - a}$
Q1MCQSum of angles
The sum of the angles of a triangle is:
- A
  $\pi$
- B
  $\dfrac{\pi}{2}$
- C
  $2\pi$
- D
  $\dfrac{3\pi}{2}$
Sum of angles
$A + B + C = π$
Q1MCQArea 2R form
The area equals $2R^{2}\sin A\sin B\sin C$; this uses:
- A
  $a=2R\sin A$ in $\tfrac12 ab\sin C$
- B
  Heron's formula
- C
  the inradius
- D
  the median
Area (2R form)
$Δ = 2 R^{2} sin A sin B sin C$
Q1MCQa and R
The side $a$ in terms of $R$ is:
- A
  $2R\sin A$
- B
  $R\sin A$
- C
  $\dfrac{R}{\sin A}$
- D
  $2R\cos A$
a in terms of R
$a = 2 R sin A$
Q1MCQIncentre distance
The distance of the incentre from vertex A is:
- A
  $\dfrac{r}{\sin(A/2)}$
- B
  $r\sin(A/2)$
- C
  $\dfrac{r}{\cos(A/2)}$
- D
  $R$
Distance of incentre from vertex
$A I = \frac{r}{sin ( A /2 )}$
Q1MCQMedian
The length of the median to side $a$ is:
- A
  $\tfrac12\sqrt{2b^{2}+2c^{2}-a^{2}}$
- B
  $\tfrac12\sqrt{b^{2}+c^{2}}$
- C
  $\sqrt{2b^{2}+2c^{2}}$
- D
  $\tfrac12(b+c)$
Length of median
$m_{a} = \frac{1}{2} 2 b^{2} + 2 c^{2} - a^{2}$

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Properties AND Solution OF Triangles formulas

Properties AND Solution OF Triangles, every formula

Sine rule

Cosine rule

Projection formula

Area (two sides + angle)

Area (Heron)

Semi-perimeter

Circumradius

Inradius

Tangent half-angle

Sine half-angle

Cosine half-angle

Napier's analogy

Area (R form)

m-n theorem

Escribed radius r₁

Sum of angles

Area (2R form)

a in terms of R

Distance of incentre from vertex

Length of median

More JEE Advanced Mathematics formula sheets

Other ways to revise this chapter

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