Question

Check whether the given points are collinear.
d(L M) = 4, d(M N) =12, d(N L) = 8.

Answer 1

Hint : In this question, we will use the concept that if a whole is divided into two parts then summation of the parts will give the original number. According to the question, L,M and N are three points. We will check whether the sum of two distances is equal to third or not.

Complete step by step solution :
In this question, we have been given three points and corresponding distances between any two points and we have to show that these points are collinear.
So, we will use the concept that if the given three points are collinear then distance between the two extreme points is equal to the sum of distance of middle points from the two extremes.
It is given the distances as:
d(L M) = 4, d(M N) =12, d(N L) = 8.
d(L M) means distance between points L and M.
d(N L) means distance between points N and L.
d(M N) means distance between points M and N.
Adding the two smaller distances ,we get:
d(L M)+ d(N L) =4+8 = 12 = d(M N).
Since the distance between M and N is largest,it means M and N must be extreme points.
This can be represented pictorially as follow:

So, we can say that points M, L and N are collinear.

Note : In question related to proof of the collinearity of three points, the information in the question can be given in two ways. In the first case, the distance between each of two points is given . In this case, we will follow the solution of this question. The question may be asked in another way where coordinates of each point is given. In this case we will show that the area of the triangle formed by these points is zero.