Question : $\triangle$LMN is right angled at M. If $\angle$N = 45$^\circ$, what is the length of MN (in cm), if NL = 9$\sqrt2$cm?
Option 1: $9\sqrt2$
Option 2: $\frac{9}{\sqrt2}$
Option 3: $18$
Option 4: $9$
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Correct Answer: $9$
Solution :
$\triangle$LMN is right angled at M.
$\angle$M = 90$^\circ$
$\angle$N = 45$^\circ$
$\angle$L = 180$^\circ$ – (45$^\circ$ + 90$^\circ$) = 45$^\circ$
So, $\triangle$LMN is an isosceles right-angled triangle.
⇒ LM = MN
Also, NL = 9$\sqrt2$ cm
Using the Pythagoras theorem,
LM$^2$ + MN$^2$ = LN$^2$
⇒ 2MN$^2$ = (9$\sqrt2$)$^2$
⇒ MN = 9 cm
Hence, the correct answer is $9$.
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