Question

In ∆ABC, the sides are 6cm, 10cm and 14cm. show that the triangle is obtuse angled with the obtuse angle to 120°.
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Answer 1

$\Huge\bigstar\:\:\tt\underline\red{QUESTION}$

In ∆ABC, the sides are 6cm, 10cm and 14cm. show that the triangle is obtuse angled with the obtuse angle to 120°.

$\Huge\bigstar\:\:\tt\underline\red{DIAGRAM}$

$\setlength{\unitlength}{1cm}\begin{picture}(6,8)\linethickness{0.075mm}\put(1, .5){\line(2, 1){3}}\put(4, 2){\line(-2, 1){2}}\put(2, 3){\line(-2, -5){1}}\put(.7, .3){ B }\put(4.05, 1.9){ A }\put(1.7, 2.95){ C }\put(3.2, 2.5){ 10 cm }\put(0.6,1.7){ 6 cm }\put(2.7, 1.05){ 14 cm }\end{picture}$

• AB = 14 cm
• AC = 10 cm
• BC = 6 cm

Here AB is the longest side. Therefore ∠C is obtuse angle

$\Huge\bigstar\:\:\tt\underline\red{SOLUTION}$

$\sf\underline\pink{By\:using\:COS\: formula\:for\:\angle C}$

$\Large\purple{\longrightarrow\:\:}$$\large\sf\purple{c^{2} \:=\:a^{2}\:+\:b^{2}\:-\:2ab\:cos(c)}$

$\Large\green{\longrightarrow\:\:}$$\large\sf\green{2ab\:cos(c)\:=\:a^{2}\:+\:b^{2}\:-\:c^{2}}$

$\Large\purple{\longrightarrow\:\:}$$\large\sf\purple{cos(c)\:=\:\frac{a^{2}\:+\:b^{2}\:-\:c^{2}}{2ab}}$

$\Large\green{\longrightarrow\:\:}$$\large\sf\green{cos(c)\:=\:\frac{(6)^{2}\:+\:(10)^{2}\:-\:(14)^{2}}{2\:×\:6\:×10}}$

$\Large\purple{\longrightarrow\:\:}$$\large\sf\purple{cos(c)\:=\:\frac{36\:+\:100\:-\:196}{120}}$

$\Large\green{\longrightarrow\:\:}$$\large\sf\green{cos(c)\:=\:\frac{-60}{120}}$

$\Large\purple{\longrightarrow\:\:}$$\large\sf\purple{cos(c)\:=\:\frac{-1}{2}}$

$\Large\green{\longrightarrow\:\:}$$\large\sf\green{c\:=\:\frac{1}{cos}×\frac{-1}{2}}$

$\Large\purple{\longrightarrow\:\:}$$\large\sf\purple{c\:=\:cos^{-1}\:(\frac{-1}{2})}$

$\Large\green{\longrightarrow\:\:}$$\large\sf\green{c\:=\:\pi\:-\:\frac{\pi}{3}}$

$\Large\purple{\longrightarrow\:\:}$$\large\sf\purple{c\:=\:\frac{(3\pi\:-\:\pi)}{3}}$

$\Large\green{\longrightarrow\:\:}$$\large\sf\green{c\:=\:\frac{2\pi}{3}}$

$\Large\purple{\longrightarrow\:\:}$$\Large\sf\purple{c\:=\:\frac{2\:×\:180}{3}}$

$\Large\green{\longrightarrow\:\:}$$\Large\sf\green{c\:=\:2\:×\:60}$

$\Large\orange{\longrightarrow\:\:}$$\Large\sf\orange{\angle C\:=\:120°}$

$\Large\star\:$ $\sf\underline\pink{It\:Will\:Be\: Obtuse\:Angled\: Triangle\:of\:120°}$

Answer 2

$\large\underline\mathrm\red{Given:-}$

• a = 6
• b = 10
• c = 14

$\large\underline\mathrm\pink{Long \: c \: will \: be \: largest \: obtuse.}$

$\large\underline\mathrm\red{Solution}$

$\implies$cos c = a² + b² - c² / 2ab

$\implies$36 + 100 - 196 / 2 × 6 × 10

$\implies$- 60/ 120

$\implies$- 1/2

$\implies$cos c = -1/2

$\implies$c = cos^-1 (-1/2)

$\implies$π - π/3

$\implies$2π/3

$\implies$2 × 180/3

$\implies$120°

$\large\underline\mathrm\red{c \: = \: 120°}$

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