Advertisements
Advertisements
Question
If G be the centroid of a triangle ABC, prove that:
AB2 + BC2 + CA2 = 3 (GA2 + GB2 + GC2)
Advertisements
Solution
Let A(x1,y1); B(x2,y2); C(x3,y3) be the coordinates of the vertices of ΔABC.Let us assume that centroid of the ΔABC is at the origin G.So, the coordinates of G are G(0,0).
Now,`(x_1+x_2+x_3)/3 =0; (y_1+y_2+y_3)/3 =0`
So, `x_1+x_2+x_3=0` ...........(1)
`y_1+y_2+y_3=0` ..........(2)
Squaring (1) and (2), we get
`x_1^2+x_2^2+x_3^2+2x_1x_2+2x_2x_3+2x_3x_1=0` ..........(3)
`y_1^2+y_2^2+y_3^2+2y_1y_2+2y_2y_3+2y_3y_1=0 ` ..........(4)
`LHS=AB^2+BC^2+CA^2`
`=[sqrt((x_2-x_1)^2 +(y_2-y_1)^2]]^2 +[sqrt((x_3-x_2)^2+(y_3-y_2)^2)]^2 +[sqrt((x_3-x_1)^2+(y_3-y_1)^2)]^2 `
`=(x_2-x_1)^2 +(y_2-y_1)^2+(x_3-x_2)^2+(y_3-y_2)^2+(x_3-x_1)^2+(y_3-y_1)^2`
`=x_1^2x_2^2-2x_1^2+y_1^2+y_2^2-2y_1y_2+x_2^2+x_3^2-2x_2x_3+y_2^2+y_2^2+y_3^2-2y_2y_3+x_1^2+x_3^2-2x_1x_3+y_1^2+y_3^2-2y_1v_3`
`=2(x_1^2+x_2^2+x_3^2)+2(y_1^2+y_2^2+y_3^2)-(2x_1x_2+2x_2x_3+2x_3x_1)-(2y_1y_2+2y_2y_3+2y_3y_1)`
`=2(x_1^2+x_2^2+x_3^2)+2(y_1^2+y_2^2+y_3^2)+(x_1^2+x_2^2+x_3^2)+(y_1^2+y_2^2+y_3^2)`
`=3(x_1^2+x_2^2+x_3^2+y_1^2+y_2^2+y_3^2)`
`RHS =3(GA^2+GB^2+GC^2)`
`=[{sqrt((x_1-0)^2+(y_1-0)^2)}^2 +{sqrt((x_2-0)^2+(y_2-0)^2)}^2 +{sqrt((x_3-0)^2+(y_3-0)^2)}^2]`
`=3[x_1^2+x_2^2+x_3^2+y_1^2+y_2^2+y_3^2]`
Hence, `AB^2+BC^2+CA^2=3(GA^2+GB^2+GC^2)`
APPEARS IN
RELATED QUESTIONS
Which point on the y-axis is equidistant from (2, 3) and (−4, 1)?
Find the coordinates of the point which divides the line segment joining (−1,3) and (4, −7) internally in the ratio 3 : 4
Prove that the points (3, -2), (4, 0), (6, -3) and (5, -5) are the vertices of a parallelogram.
The line joining the points (2, 1) and (5, −8) is trisected at the points P and Q. If point P lies on the line 2x − y + k = 0. Find the value of k.
Find the co-ordinates of the point equidistant from three given points A(5,3), B(5, -5) and C(1,- 5).
Find the coordinates of the midpoints of the line segment joining
P(-11,-8) and Q(8,-2)
In what ratio does y-axis divide the line segment joining the points (-4, 7) and (3, -7)?
Find the ratio which the line segment joining the pints A(3, -3) and B(-2,7) is divided by x -axis Also, find the point of division.
The base BC of an equilateral triangle ABC lies on y-axis. The coordinates of point C are (0, −3). The origin is the midpoint of the base. Find the coordinates of the points A and B. Also, find the coordinates of another point D such that ABCD is a rhombus.
Find the area of a quadrilateral ABCD whose vertices area A(3, -1), B(9, -5) C(14, 0) and D(9, 19).
If the point A(0,2) is equidistant from the points B(3,p) and C(p, 5), find p.
The co-ordinates of point A and B are 4 and -8 respectively. Find d(A, B).
The ordinate of any point on x-axis is
If (0, −3) and (0, 3) are the two vertices of an equilateral triangle, find the coordinates of its third vertex.
If \[D\left( - \frac{1}{5}, \frac{5}{2} \right), E(7, 3) \text{ and } F\left( \frac{7}{2}, \frac{7}{2} \right)\] are the mid-points of sides of \[∆ ABC\] , find the area of \[∆ ABC\] .
If A (1, 2) B (4, 3) and C (6, 6) are the three vertices of a parallelogram ABCD, find the coordinates of fourth vertex D.
If the centroid of the triangle formed by the points (a, b), (b, c) and (c, a) is at the origin, then a3 + b3 + c3 =
The coordinates of a point on x-axis which lies on the perpendicular bisector of the line segment joining the points (7, 6) and (−3, 4) are
A line intersects the y-axis and x-axis at P and Q , respectively. If (2,-5) is the mid-point of PQ, then the coordinates of P and Q are, respectively
Seg AB is parallel to X-axis and coordinates of the point A are (1, 3), then the coordinates of the point B can be ______.
