Question

find the value of ∆ABC​

Answer 1

$\underline{\underline{\sf{\pink{Given :} }}}$

• ∆ABC is a right Angled triangle
• AB = AC
• AD is bisector of ∠A

$\sf AD = 2\sqrt{2}$

⠀ ⠀⠀ ⠀ ⠀⠀ ⠀

$\underline{\underline{\sf{\pink{To\:Find :} }}}$

• Perimeter of ∆ABC

⠀ ⠀⠀ ⠀ ⠀⠀ ⠀

$\underline{\underline{\sf{\pink{Solution :} }}}$

• In ∆ABC,

• AB = AC

$\underline{\tt{As\:we\:know\:that,}}$

$\star\:{\boxed{\sf{\pink{Angle \: opposite \: to \: equal \: sides \: are \: equal }}}}$

$: \: \implies \sf{\angle C = \angle B - (i)}$

$\underline{\tt{Now, \:by \: angle\:sum \: property,}}$

∠A + ∠B + ∠C = 180°

From (i)

∠A + ∠B + ∠B = 180°

$: \: \implies \sf{90^{\circ} + 2 \angle B = 180^{\circ}}$

$: \: \implies \sf{2 \angle B = 180^{\circ} - 90^{\circ}}$

$: \: \implies \sf{2 \angle B = 90^{\circ}}$

$: \: \implies \sf{ \angle B = \dfrac{90^{\circ}}{2}}$

$: \: \implies \sf{\angle B = 45^{\circ}}$

$\underline{\tt{Now, \: \angle B = \angle C}}$

$: \: \implies \sf{\angle C = 45^{\circ}}$

_______________________________

In ∆ABD

As, AD bisects BC,

Therefore, ∠ADB = 90°

$\underline{\tt{Now, \: by \: angle\:sum \: property,}}$

• ∠B + ∠DAB + ∠ADB = 180°
• 45° + ∠DAB + 90° = 180°
• 45° + ∠DAB = 180° - 90°
• ∠DAB = 90° - 45°
• ∠DAB = 45°

As, ∠B = ∠DAB (both 45°) and we know that sides opposite to equal angles are equal

Therefore, BD = AD - (ii)

_______________________________

In ∆ACD

As, AD bisects BC,

Therefore, ∠ADC = 90°

$\underline{\tt{Now, \: by \: angle\:sum \: property,}}$

• ∠C + ∠DAC + ∠ADC = 180°
• 45° + ∠DAC + 90° = 180°
• 45° + ∠DAC = 180° - 90°
• ∠DAC = 90° - 45°
• ∠DAC = 45°

As, ∠C = ∠DAC (both 45°) and we know that sides opposite to equal angles are equal

Therefore, CD = AD -(iii)

_______________________________

From (ii) and (iii),

BD + CD = AD + AD

Now, we have BD + CD = BC

$: \: \implies \sf{BC = 2AD}$

Now, we are given AD = $\sf 2\sqrt{2}cm$

$: \: \implies \sf{BC = 2 \times 2\sqrt{2}}$

$: \: \implies \sf{BC = 4\sqrt{2}cm }$

_______________________________

• In ∆ABC

$\underline{\tt{By \: Pythagoras' \: theorem,}}$

$\star\:{\boxed{\sf{\pink{H^{2} = B^{2} + P^{2}}}}}$

$: \: \implies \sf{BC^{2} = AB^{2} + AC^{2}}$

$: \: \implies \sf{{(4\sqrt{2})}^{2} = AB^{2} + AC^{2}}$

$\underline{\tt{Now, \: we \: are \: given \: AB = AC}}$

$: \: \implies \sf{{(4\sqrt{2})}^{2} = AB^{2} + AB^{2}}$

$: \: \implies \sf{16 \times 2 = 2AB^{2}}$

$: \: \implies \sf{32 = 2AB^{2}}$

$: \: \implies \sf{\dfrac{32}{2} = AB^{2}}$

$: \: \implies \sf{16 = AB^{2}}$

$: \: \implies \sf{\sqrt{16} = AB}$

$: \: \implies \sf{AB = 4 cm}$

As, AB = AC (given)

$: \: \implies \sf{AC = 4 cm}$

Now, We have

AB = 4 cm

AC = 4 cm

$\sf{BC = 4\sqrt{2}cm }$

_______________________________

$\underline{\tt{Now, \:we \: have \: to \: find \: perimeter,}}$

$\star\:{\boxed{\sf{\pink{Perimeter \: of \: triangle = a + b + c}}}}$

• In ∆ABC,
• a = AB = 4 cm
• b = AC = 4 cm
• c = $\sf{BC = 4\sqrt{2}cm }$

$: \implies \sf{Perimeter = 4 cm + 4 cm + 4\sqrt{2}cm }$

$: \implies \sf{Perimeter = 8 + 4\sqrt{2}cm}$

$\pink{\sf \therefore \: Perimeter \: of \: \triangle ABC = 8 + 4\sqrt{2}cm}$

Answer 2

$\huge{\underline{\underline{\boxed{\sf{\purple{αɳรᏇɛƦ \: ࿐}}}}}}$

Since the French Revolution, liberalism had stood for the end of autocracy and clerical privileges, a constitution and representative government through parliament. Nineteenth-century liberals also stressed the inviolability of private property. The memory of the French Revolution nonetheless continued to inspire liberals. One of the major issues taken up by the liberal-nationalists, who criticised the new conservative order, was freedom of the press.

Parallel to the revolts of the poor, unemployed and starving peasants and workers in many European countries in the year 1848, a revolution led by the educated middle classes was under way. Events of February 1848 in France had brought about the abdication of the monarch and a republic based on universal male suffrage had been proclaimed. In other parts of Europe where independent nation-states did not yet exist – such as Germany, Italy, Poland, the Austro-Hungarian Empire – men and women of the liberal middle classes combined their demands for constitutionalism with national unification. They took advantage of the growing popular unrest to push their demands for the creation of a nation-state on parliamentary principles – a constitution, freedom of the press and freedom of association.Nationalist feelings were widespread among middle-class Germans, who in 1848 tried to unite the different regions of the German confederation into a nation-state governed by an elected parliament. This liberal initiative to nation-building was, however, repressed by the combined forces of the monarchy and the military, supported by the large landowners (called Junkers) of Prussia. From then on, Prussia took on the leadership of the movement for national unification. Its chief minister, Otto von Bismarck, was the architect of this process carried out with the help of the Prussian army and bureaucracy. Three wars over seven years – with Austria, Denmark and France – ended in Prussian victory and completed the process of unification. In January 1871, the Prussian king, William I, was proclaimed German Emperor in a ceremony held at Versailles.Since the French Revolution, liberalism had stood for the end of autocracy and clerical privileges, a constitution and representative government through parliament. Nineteenth-century liberals also stressed the inviolability of private property. The memory of the French Revolution nonetheless continued to inspire liberals. One of the major issues taken up by the liberal-nationalists, who criticised the new conservative order, was freedom of the press.

Parallel to the revolts of the poor, unemployed and starving peasants and workers in many European countries in the year 1848, a revolution led by the educated middle classes was under way. Events of February 1848 in France had brought about the abdication of the monarch and a republic based on universal male suffrage had been proclaimed. In other parts of Europe where independent nation-states did not yet exist – such as Germany, Italy, Poland, the Austro-Hungarian Empire – men and women of the liberal middle classes combined their demands for constitutionalism with national unification. They took advantage of the growing popular unrest to push their demands for the creation of a nation-state on parliamentary principles – a constitution, freedom of the press and freedom of association.