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# In the figure, if $\dfrac{{NT}}{{AB}} = \dfrac{9}{5}$ , and $MB = 10$ , then $MN$ will be$A) 18cm \\B) 20cm \\C) 30cm \\D) 90cm \\$

Last updated date: 13th Jun 2024
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Hint: In this question, first of all observe the diagram relating with the given information in the question to get the idea about what has been asked for as an answer & how we have to proceed stepwise. For solving the question, just use similarity property for triangles which are concerned with the given sides & angles. Have a look at an angle given of the triangle and according to the relation ratio of corresponding sides are proportional to get the answer.
Given information is $NT:AB = 9:5$.

In $\Delta AMB$ & $\Delta NMT$-
$\angle MAB = \angle MTN$- (by given figure)
$\angle AMB = \angle NMT = {30^0}$ (common angle to both triangles)
$\angle MNT = \angle MBA$ (by given figure)
$\therefore \Delta AMB \sim \Delta NMT$ i.e, $\Delta AMB$ Hence $\Delta AMB$ is congruent to $\vartriangle NMT$.
$\therefore$Hence , the two triangles $\Delta AMB$ & $\vartriangle NMT$ are similar.
As $\dfrac{{AB}}{{NT}} = \dfrac{{MB}}{{MN}}$ & $MB = 10cm$ & $NT:AB = 9:5$, putting the values of
$\therefore \dfrac{5}{9} = \dfrac{{10}}{{MN}}$
$\Rightarrow MN = 18cm$
$\therefore$ Length of $MN = 18cm$