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### High School Mathematics - 29.16 Centroid of a Triangle

 To find the centroid of a triangle whose vertices are given Let A(x1, y1), B(x2, y2) and C(x3, y3) be the vertices of the triangle ABC. Let AD be the median bisecting the base. Then D = {(x2 + x3)/2, (y2 + y3)/2} Now the point G on AD, which divides it internally in the ratio 2:1, is the centroid. If (x,y) are the coordinates of G, then x = 2x(x2+x3)/2 + (1 x x1)/2+1 = (x1+x2+x3)/3 y = 2x(y2+y3)/2 + (1 x y1)/(2+1) = (y1+y2+y3)/3 Hence, the coordinates of the centroid are given by x = (x1+x2+x3)/3, y = (y1+y2+y3)/3. Directions: Solve the following.
 Q 1: The coordnates of the vertices of the triangle are (4, -3), (-5,2) and (x,y). If the centroid of the triangle is at the origin, what is the value of x and y.(0,1)10(1,1) Q 2: Find the length of the median AD of a triangle whose vertices are A(4,4), B(3,-2) and C(-3,16).37510 Q 3: Find the centroid of the triangle formed by the lines 3x-y-11 = 0, 7y+x-11 = 0, 2x+3y = 0(-4/3, -1/3)(-4/3, 1/3)(4/3, 1/3) Q 4: Origin is the centroid of the triangle, whose two vertices are (5, -1), (-2,3).Find the third vertex.(-3, -2)(3,-2)(3, 2) Q 5: The vertices of a triangle are (1,2), (h, -3) and (-4, k). FInd the values of h and k if the centroid of the triangle be at the point (5, -1).(-18, -2)(2, 18)(8, 2)(18, -2) Q 6: If (2,4) is the centroid of the triangle two of whose vertices are (7,1) and (2,7). Find the remaining vertex.(-3,4)(-3,-4)(3,4)(3,-4) Q 7: Find the centroid of the triangle whose sides are formed by the lines 2x-3y = 1, y + 1 = 0 and 4x - 5y = 1.(0,0)(1/3, 0)(0, 1/3)(1/3, 1/3) Q 8: Find the third vertex of a triangle if two of its vertices are at (-1,4) and (5,2)(-4, -15)(4, -15)(4, 15)(-4, 15) Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!