Coordinates of the mid-point M (x,y) of the segment AB are obtained by taking in the section formula:
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Illustration: Find the distance of the point (1,2) from the mid-point of the line segment joining the points (6,8) and (2,4).
Solution: Let the points (1,2), (6,8), and (2,4) be denoted by A, B, and C respectively. Let M be the mid-point of BC. Coordinates of M, the mid-point of BC, are given by
We have
Illustration: Find the lengths of medians of the triangle with vertices A(2,2), B(0,2), and C(2,-4).
Solution: Let the coordinates of A, B, and C be (2, 2), (0,2), and (2, -4) respectively. Let D, E, F be the mid-points of BC, CA, and AB respectively.
Then the coordinates of D, E, F are given by
, ,
or D (1, -1) E ( 2, -1) , F (1, 2)
Length of median AD=
Length of median BE=
Length of median CF =
Lengths of medians are .
Illustration: Three consecutive vertices of a parallelogram are A(1,2), B(1,0) C(4,0). Find the fourth vertex D.
Solution: Let coordinates of D be (x,y). As ABCD is a parallelogram, the diagonals AC and BD bisect each other. If M is the mid-point of AC, then coordinates of M are
As M is the mid-point of BD also, coordinates of M are
We have
Coordinates of D are (4,2)
The centroid of a triangle is the point of concurrence of the medians of a triangle.
If G is the centroid of triangle ABC, then G divides the median AD in the ratio 2:1.
Let D be the mid-point of BC, the coordinates of D are
As G(x,y) divides AD in the ratio 2 : 1
and
Thus, the coordinates of the centroid are
Illustration: Find the third vertex of the triangle whose two vertices are (-3,1) and (0,-2) and the centroid is the origin.
Solution: Let the two given vertices be A(-3,1) and B(0,-2). Let the third vertex be C(x,y) . Coordinates of the centroid of is given by
.
As the centroid of is given to be the origin,
we have
Thus, the coordinates of the third vertex are (3,1).
If is the mid point of the segment joining the points Q (-6,5) and R (-2,3) . Then the value of a is : | |||
Right Option : D | |||
View Explanation |
The co-ordinates of the mid-point of the line joining points (2a+1,4) and (5,b-1) are (2a,b). The value of (a+b) is ________________ | |||
Right Option : B | |||
View Explanation |
C is the mid-point of PQ, if P is (4,x), C is (y,-1) and Q is (-2,4), then x and y respectively are | |||
Right Option : A | |||
View Explanation |
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