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If A (-1, 3), B (1, -1) And C (5, 1) Are the Vertices of a Triangle Abc, Find the Length of the Median Through A. - CBSE Class 10 - Mathematics

Question

If A (-1, 3), B (1, -1) and C (5, 1) are the vertices of a triangle ABC, find the length of the median through A.

Solution

The distance d between two points (x_1,y_1) and (x_2, y_2) is given by the formula

d = sqrt((x_1-x_2)^2 + (y_1 - y_2)^2)

The co-ordinates of the midpoint (x_m,y_m) between two points (x_1, y_1) and (x_2, y_2) is given by,

(x_m,y_m) = (((x_1 + x_2)/2)"," ((y_1+y_2)/2))

Here, it is given that the three vertices of a triangle are A(−1,3), B(1,−1) and C(5,1).

The median of a triangle is the line joining a vertex of a triangle to the mid-point of the side opposite this vertex.

Let ‘D’ be the mid-point of the side ‘BC’.

Let us now find its co-ordinates.

(x_D,y_D) = (((1 + 5)/2)"," ((-1+1)/2))

(x_D, y_D) = (3,0)

Thus we have the co-ordinates of the point as D(3,0).

Now, let us find the length of the median ‘AD’.

AD = sqrt((-1-3)^2 + (3 -  0)^2)

= sqrt((-4)^2 + (3)^2)

= sqrt(16 + 9)