Question

PLEASE ANSWER !!!!!
Let ABCD is a square of each side of length 2 cms. M is the midpoint of AB
and ‘P’ is a variable point on BC. Find the smallest possible value of
DP + PM.

Answer 1

Answer:

I assumed the coordinates of P = (h,2) to get the value of DP+PM= (h−2)2+4−−−−−−−−−−√+h2+1−−−−−√. Then I differentiated the equation wrt to h to get: h(h2+1−−−−−√)−2h2+1−−−−−√+hh2+8−4h−−−−−−−−−√. On equating this expression with 0, I will probably get the answer.

The easiest way is to use a reflection. Let N be the symmetric of M with respect to B:

Then DP+PM=DP+PN≥DN, and the minumum for DP+PM is achieved when P=BC∩DN.

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