^@ABCD^@ is a square with each side measuring ^@144 \space cm^@. | Math Question and Answer

Answer:

$180 \space cm$

Step by Step Explanation:

Given, $BC = 144 \space cm$ and $CM = 36 \space cm$
$\implies BM = CB - CM = 144 - 36 = 108$
Let's join $A$ to $N$
$\begin{align} & \text{Since } \triangle ADN \cong \triangle CDN && [\text{By SAS criterion}] \\ & \therefore AN = CN && [\text{Corresponding sides of congruent triangles}] \space\space\space\space \end{align}$
$\implies \space AN + MN = CN + NM$
Observe that the value of $AN + NM$ is least when $ANM$ is a straight line.

Now, if

$ANM$ is a straight line, then

$\triangle AMB$ is a right-angled triangle.

$\therefore$ by Pythagoras theorem,

Least value of

$AN + NM$

$\begin{align} & = \sqrt{ AB^2 + BM^2 } \\ & = \sqrt{ 144^2 + 108^2 } \\ & = 180 \end{align}$

From step 2, we have

$AN + MN = CN + NM$ .

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