Solultion of Triangle - Notes, Concept and All Important Formula

PROPERTIES AND SOLUTIONS OF TRIANGLE

1. SINE FORMULAE :

In any triangle $\mathrm{ABC}$
$\dfrac{\mathrm{a}}{\sin \mathrm{A}}=\dfrac{\mathrm{b}}{\sin \mathrm{B}}=\dfrac{\mathrm{c}}{\sin \mathrm{C}}=\lambda=\dfrac{\mathrm{abc}}{2 \Delta}=2 \mathrm{R}$
where $\mathrm{R}$ is circumradius and $\Delta$ is area of triangle.

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All Chapter Notes, Concept and Important Formula

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2. COSINE FORMULAE :

(a) $\mathrm{\cos A=\dfrac{b^{2}+c^{2}-a^{2}}{2 b c}}$ or $\mathrm{a^{2}=b^{2}+c^{2}-2 b c \cos A}$
(b) $\mathrm{\cos B=\dfrac{c^{2}+a^{2}-b^{2}}{2 c a}}$
(c) $\mathrm{\cos C=\dfrac{a^{2}+b^{2}-c^{2}}{2 a b}}$

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3. PROJECTION FORMULAE :

(a) $b \cos C+c \cos B=a$
(b) $c \cos A+a \cos C=b$
(c) $a \cos B+b \cos A=c$

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4. NAPIER'S ANALOGY (TANGENT RULE) :

(a) $\tan \left(\dfrac{\mathrm{B}-\mathrm{C}}{2}\right)=\dfrac{\mathrm{b}-\mathrm{c}}{\mathrm{b}+\mathrm{c}} \cot \dfrac{\mathrm{A}}{2}$

(b) $\tan \left(\dfrac{\mathrm{C}-\mathrm{A}}{2}\right)=\dfrac{\mathrm{c}-\mathrm{a}}{\mathrm{c}+\mathrm{a}} \cot \dfrac{\mathrm{B}}{2}$

(c) $\tan \left(\dfrac{A-B}{2}\right)=\dfrac{a-b}{a+b} \cot \dfrac{C}{2}$

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5. HALF ANGLE FORMULAE :

$s=\dfrac{a+b+c}{2}=$ semi-perimeter of triangle.

(a) (i) $\sin \dfrac{A}{2}=\sqrt{\dfrac{(s-b)(s-c)}{b c}}$
(ii) $\sin \dfrac{B}{2}=\sqrt{\dfrac{(s-c)(s-a)}{c a}}$
(iii) $\sin \dfrac{C}{2}=\sqrt{\dfrac{(s-a)(s-b)}{a b}}$

(b) (i) $\cos \dfrac{A}{2}=\sqrt{\dfrac{s(s-a)}{b c}}$
(ii) $\cos \dfrac{B}{2}=\sqrt{\dfrac{s(s-b)}{c a}}$
(iii) $\cos \dfrac{C}{2}=\sqrt{\dfrac{s(s-c)}{a b}}$

(c) (i) $\tan \dfrac{A}{2}=\sqrt{\dfrac{(s-b)(s-c)}{s(s-a)}}=\dfrac{\Delta}{s(s-a)}$
(ii) $\tan \dfrac{B}{2}=\sqrt{\dfrac{(s-c)(s-a)}{s(s-b)}}=\dfrac{\Delta}{s(s-b)}$
(iii) $\tan \dfrac{C}{2}=\sqrt{\dfrac{(s-a)(s-b)}{s(s-c)}}=\dfrac{\Delta}{s(s-c)}$

(d) Area of Triangle $\Delta=\sqrt{s(s-a)(s-b)(s-c)}$
$=\dfrac{1}{2} b c \sin A=\dfrac{1}{2} c a \sin B=\dfrac{1}{2} a b \sin C$
$=\dfrac{1}{4} \sqrt{2\left(a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}\right)-a^{4}-b^{4}-c^{4}}$

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6. RADIUS OF THE CIRCUMCIRCLE 'R' :

Circumcentre is the point of intersection of perpendicular bisectors of the sides and distance between circumcentre & vertex of triangle is called circumradius 'R'.

$\mathrm{R}=\dfrac{\mathrm{a}}{2 \sin \mathrm{A}}=\dfrac{\mathrm{b}}{2 \sin \mathrm{B}}=\dfrac{\mathrm{c}}{2 \sin \mathrm{C}}=\dfrac{\mathrm{abc}}{4 \Delta}$.

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7. RADIUS OF THE INCIRCLE 'r' :

Point of intersection of internal angle bisectors is incentre and perpendicular distance of incentre from any side is called inradius 'r'.

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8. RADII OF THE EX-CIRCLES :

Point of intersection of two external angle and one internal angle bisectors is excentre and perpendicular distance of excentre from any side is called exradius. If $r_{1}$ is the radius of escribed circle opposite to angle $A$ of $\triangle A B C$ and so on then :

(a) $\mathrm{r}_{1}=\dfrac{\Delta}{\mathrm{s}-\mathrm{a}}=\mathrm{s} \tan \dfrac{\mathrm{A}}{2}=4 \mathrm{R} \sin \dfrac{\mathrm{A}}{2} \cos \dfrac{\mathrm{B}}{2} \cos \dfrac{\mathrm{C}}{2}$

(b) $\mathrm{r}_{2}=\dfrac{\Delta}{\mathrm{s}-\mathrm{b}}=\operatorname{stan} \dfrac{\mathrm{B}}{2}=4 \mathrm{R} \cos \dfrac{\mathrm{A}}{2} \sin \dfrac{\mathrm{B}}{2} \cos \dfrac{\mathrm{C}}{2}$

(c) $\mathrm{r}_{3}=\dfrac{\Delta}{\mathrm{s}-\mathrm{c}}=\operatorname{stan} \dfrac{\mathrm{C}}{2}=4 \mathrm{R} \cos \dfrac{\mathrm{A}}{2} \cos \dfrac{\mathrm{B}}{2} \sin \dfrac{\mathrm{C}}{2}$

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9. LENGTH OF ANGLE BISECTOR, MEDIANS & ALTITUDE :

If $m_{a}, \beta_{a}$ & $h_{a}$ are the lengths of a median, an angle bisector & altitude from the angle A then,
$\dfrac{1}{2} \sqrt{b^{2}+c^{2}+2 b c \cos A}=m_{a}=\dfrac{1}{2} \sqrt{2 b^{2}+2 c^{2}-a^{2}}$
and $\quad \beta_{a}=\dfrac{2 b c \cos \dfrac{A}{2}}{b+c}, h_{a}=\dfrac{a}{\cot B+\cot C}$
Note that $m_{a}^{2}+m_{b}^{2}+m_{c}^{2}=\dfrac{3}{4}\left(a^{2}+b^{2}+c^{2}\right)$

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10. ORTHOCENTRE AND ORTHIC TRIANGLE 

(a) Point of intersection of altitudes is orthocentre & the triangle KLM which is formed by joining the feet of the altitudes is called the orthic triangle.

(b) The distances of the orthocentre from the angular points of the $\Delta \mathrm{ABC}$ are $2 \mathrm{R} \cos \mathrm{A}, 2 \mathrm{R} \cos \mathrm{B},$ & $2 \mathrm{R} \cos \mathrm{C}$

(c) The distance of orthocentre from sides are $2 \mathrm{R} \cos \mathrm{B} \cos \mathrm{C}$, $2 \mathrm{R} \cos C \cos A$ and $2 \mathrm{R} \cos A \cos B$

(d) The sides of the orthic triangle are a $\cos A(=R \sin 2 A)$, $b \cos B(=R \sin 2 B)$ and $c \cos C(=R \sin 2 C)$ and its angles are $\pi-2 A, \pi-2 B$ and $\pi-2 C$

(e) Circumradii of the triangles $\mathrm{PBC}, \mathrm{PCA}, \mathrm{PAB}$ and $\mathrm{ABC}$ are equal.

(f) Area of orthic triangle $=2 \Delta \cos A \cos B \cos C$ $=\dfrac{1}{2} \mathrm{R}^{2} \sin 2 \mathrm{~A} \sin 2 \mathrm{~B} \sin 2 \mathrm{C}$

(g) Circumradii of orthic triangle $=\mathrm{R} / 2$

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11. EX-CENTRAL TRIANGLE :

(a) The triangle formed by joining the three excentres $\mathrm{I}_{1}, \mathrm{I}_{2}$ and $\mathrm{I}_{3}$ of $\triangle \mathrm{ABC}$ is called the excentral or excentric triangle.

(b) Incentre I of $\triangle \mathrm{ABC}$ is the orthocentre of the excentral $\Delta \mathrm{I}_{1} \mathrm{I}_{2} \mathrm{I}_{3}$.

(c) $\triangle \mathrm{ABC}$ is the orthic triangle of the $\Delta \mathrm{I}_{1} \mathrm{I}_{2} \mathrm{I}_{3}$.

(d) The sides of the excentral triangle are
$4 \mathrm{R} \cos \dfrac{\mathrm{A}}{2}, 4 \mathrm{R} \cos \dfrac{\mathrm{B}}{2}$ and $4 \mathrm{R} \cos \dfrac{\mathrm{C}}{2}$
and its angles are $\dfrac{\pi}{2}-\dfrac{A}{2}, \dfrac{\pi}{2}-\dfrac{B}{2}$ and $\dfrac{\pi}{2}-\dfrac{C}{2}$.

(e) $\mathrm{II}_{1}=4 \mathrm{R} \sin \dfrac{\mathrm{A}}{2} ; \mathrm{II}_{2}=4 \mathrm{R} \sin \dfrac{\mathrm{B}}{2} ; \mathrm{II}_{3}=4 \mathrm{R} \sin \dfrac{\mathrm{C}}{2}$.

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12. THE DISTANCES BETWEEN THE SPECIALPOINTS :

(a) The distance between circumcentre and orthocentre $=\mathrm{R} \sqrt{1-8 \cos \mathrm{A} \cos \mathrm{B} \cos \mathrm{C}}$

(b) The distance between circumcentre and incentre $=\sqrt{\mathrm{R}^{2}-2 \mathrm{Rr}}$

(c) The distance between incentre and orthocentre $\mathrm{=\sqrt{2 r^{2}-4 R^{2} \cos A \cos B \cos C}}$

(d) The distance between circumcentre & excentre is $=\mathrm{R} \sqrt{1+8 \sin \dfrac{\mathrm{A}}{2} \cos \dfrac{\mathrm{A}}{2} \cos \dfrac{\mathrm{C}}{2}}$$=\sqrt{\mathrm{R}^{2}+2 \mathrm{Rr}_1}$ & so on.

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13. m-n THEOREM :

(m + n) cot $\theta$ = m cot $\alpha$ – n cot $\beta$
(m + n) cot $\theta$ = n cot B – m cot C

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14. IMPORTANT POINTS:

(a) (i) If $a \cos B=b \cos A$, then the triangle is isosceles.
(ii) If $a \cos A=b \cos B$, then the triangle is isosceles or right angled.

(b) In Right Angle Triangle :

(i) $a^{2}+b^{2}+c^{2}=8 R^{2}$
(ii) $\cos ^{2} A+\cos ^{2} B+\cos ^{2} C=1$

(c) In equilateral triangle :

(i) $\mathrm{R}=2 \mathrm{r}$
(ii) $r_{1}=r_{2}=r_{3}=\dfrac{3 R}{2}$
(iii) $\mathrm{r}: \mathrm{R}: \mathrm{r}_{1}=1: 2: 3$
(iv) area $=\dfrac{\sqrt{3} a^{2}}{4}$
(v) $\mathrm{R}=\dfrac{\mathrm{a}}{\sqrt{3}}$

(d) (i) The circumcentre lies (1) inside an acute angled triangle, (2) outside an obtuse angled triangle & (3) at mid point of the hypotenuse of right angled triangle.
(ii) The orthocentre of right angled triangle is the vertex at the right angle.
(iii) The orthocentre, centroid & circumcentre are collinear & centroid divides the line segment joining orthocentre & circumcentre internally in the ratio $2: 1$, except in case of equilateral triangle. In equilateral triangle all these centres coincide.

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15. REGULAR POLYGON :

Consider a 'n' sided regular polygon of side length 'a'.

(a) Radius of incircle of this polygon $\mathrm{r}=\dfrac{\mathrm{a}}{2} \cot \dfrac{\pi}{\mathrm{n}}$

(b) Radius of circumcircle of this polygon $\mathrm{R}=\dfrac{\mathrm{a}}{2} \operatorname{cosec} \dfrac{\pi}{\mathrm{n}}$

(c) Perimeter & area of regular polygon
Perimeter $=\mathrm{n} \mathrm{a}=2 \mathrm{nr} \tan \dfrac{\pi}{\mathrm{n}}=2 \mathrm{nR} \sin \dfrac{\pi}{\mathrm{n}}$
Area $=\dfrac{1}{2} \mathrm{nR}^{2} \sin \dfrac{2 \pi}{\mathrm{n}}=\mathrm{nr}^{2} \tan \dfrac{\pi}{\mathrm{n}}=\dfrac{1}{4} \mathrm{na}^{2} \cot \dfrac{\pi}{\mathrm{n}}$

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16. CYCLIC QUADRILATERAL :

(a) Quadrilateral $\mathrm{ABCD}$ is cyclic if $\angle \mathrm{A}+\angle \mathrm{C}=\pi$ $=\angle \mathrm{B}+\angle \mathrm{C}$ (opposite angle are supplementary angles)

(b) Area $=\sqrt{(s-a)(s-b)(s-c)(s-d)}$, where $2 s=a+b+c+d$

(c) $\cos B=\dfrac{a^{2}+b^{2}-c^{2}-d^{2}}{2(a b+c d)}$ & similarly other angles.

(d) Ptolemy's theorem : If $A B C D$ is cyclic quadrilateral, then $\mathrm{AC} \cdot \mathrm{BD}=\mathrm{AB} \cdot \mathrm{CD}+\mathrm{BC} \cdot \mathrm{AD}$

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17. SOLUTION OF TRIANGLE :

Case-I : Three sides are given then to find out three angles use
$\cos A=\dfrac{b^{2}+c^{2}-a^{2}}{2 b c}, \cos B=\dfrac{c^{2}+a^{2}-b^{2}}{2 a c}$ & $\cos C=\dfrac{a^{2}+b^{2}-c^{2}}{2 a b}$

Case-II : Two sides & included angle are given:

Let sides $a, b$ & angle $C$ are given then use $\tan \dfrac{A-B}{2}=\dfrac{a-b}{a+b} \cot \dfrac{C}{2}$ and find value of $A-B$.....(i)

& $\dfrac{A+B}{2}=90^{\circ}-\dfrac{C}{2}$....(ii) $\quad c=\dfrac{a \sin C}{\sin A}$.....(iii)

Case-III :

Two sides $\mathrm{a}, \mathrm{b}$ $\&$ angle $\mathrm{A}$ opposite to one of them is given

(a) If $\mathrm{a}<\mathrm{b} \sin \mathrm{A} \quad$ No triangle exist
(b) If $a=b \sin A$ & $A$ is acute, then one triangle exist which is right angled.
(c) $\mathrm{a}>\mathrm{bsin} \mathrm{A}, \mathrm{a}<\mathrm{b}$ & $\mathrm{~A}$ is acute, then two triangles exist
(d) $\mathrm{a}>\mathrm{bsin} \mathrm{A}, \mathrm{a}>\mathrm{b}$ & $\mathrm{~A}$ is acute, then one triangle exist
(e) $\mathrm{a}>\mathrm{bsin} \mathrm{A}$ &  $\mathrm{~A}$ is obtuse, then there is one triangle if $\mathrm{a}>\mathrm{b}$ & no triangle if $\mathrm{a}<\mathrm{b}$.

Note : Case-III can be analysed algebraically using Cosine rule as $\cos A=\dfrac{b^{2}+c^{2}-a^{2}}{2 b c}$, which is quadratic in $c$.

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18. ANGLES OF ELEVATION AND DEPRESSION 

Let OP be a horizontal line in the vertical plane in which an object $\mathrm{R}$ is given and let OR be joined.

In Fig. (a), where the object $\mathrm{R}$ is above the horizontal line $\mathrm{OP}$, the angle POR is called the angle of elevation of the object $\mathrm{R}$ as seen from the point O. In Fig. (b) where the object $\mathrm{R}$ is below the horizontal line OP, the angle POR is called the angle of depression of the object $\mathrm{R}$ as seen from the point $\mathrm{O}$.

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