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Question

ABC is a right angled triangle in which B=90 and BC=a. If n points L1,L2,...,Ln on AB are such that AB is divided in n+1 equal parts and L1M1,L2M2,......LnMn are line segments parallel to BC and M1,M2,...,Mn are on AC then the sum of the lengths of L1M1,L2M2,......LnMn is

A
a(n+1)2
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B
a(n1)2
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C
an2
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D
impossible to find from the given data
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Solution

The correct option is D an2
Let the length of each (n+1) equal parts on AB=x
And total length of AB=b
Consider length LiMi=pi where i=1,2,...,n
Length L1A=nx=bn(n+1)
By congruity of ΔABC and ΔAL1M1,ab=p1(n+1)bn
p1=nan+1
Similarly, p2=(n1)an+1
p3=(n2)an+1


pn=an+1
p1+p2++pn=nan+1+(n1)an+1++2an+1+an+1
=an+1{n+(n1)++2+1}=an+1{n(n+1)2}
=an2
355666_132010_ans_c835e71321ce4846a9644a8eae6c7338.png

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