Question : In a triangle ABC, if $\angle B=90^{\circ}, \angle C=45^{\circ}$ and AC = 4 cm, then the value of BC is:
Option 1: $\sqrt{2} \mathrm{~cm}$
Option 2: $4 \mathrm{~cm}$
Option 3: $2 \sqrt{2} \mathrm{~cm}$
Option 4: $4 \sqrt{2} \mathrm{~cm}$
Correct Answer: $2 \sqrt{2} \mathrm{~cm}$
Solution :
AC = 4 cm
$\angle B=90^{\circ}$
$\angle C=45^{\circ}$
$\therefore \angle A = 180^{\circ}-(90^{\circ}+45^{\circ}) = 45^{\circ}$
Since $\angle A= \angle C$,
$AB = BC$ -----------(i)
By Pythagoras theorem,
$AB^2 + BC^2 = AC^2$
⇒ $BC^2 + BC^2 = 4^2$
⇒ $2BC^2 = 16$
⇒ $BC^2 = 8$
⇒ $BC = \sqrt8 = 2\sqrt2$
Hence, the correct answer is $2\sqrt2$ cm.
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