ABCD is a square with each side measuring 144 cm. M is a point on CB such that CM=36 cm. If N is a variable point on the diagonal DB, find the least value of CN+MN.
D C B A M N


Answer:

180 cm

Step by Step Explanation:
  1. Given, BC=144 cm and CM=36 cm
    BM=CBCM=14436=108
    Let's join A to N
    D C B A M N
  2. Since ADNCDN[By SAS criterion]AN=CN[Corresponding sides of congruent triangles]    
     AN+MN=CN+NM
    Observe that the value of AN+NM is least when ANM is a straight line.
  3. Now, if ANM is a straight line, then AMB is a right-angled triangle.
    by Pythagoras theorem,
    Least value of AN+NM =AB2+BM2=1442+1082=180
    From step 2, we have AN+MN=CN+NM.
  4. Hence, the least value of CN+MN is 180 cm.

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