If the points (1, -1), (2, -1) and (4, -3) are the mid-points of the sides of a triangle, then write the coordinates of its centroid.
If the points (1, -1), (2, -1) and (4, -3) are the mid-points of the sides of a triangle, then write the coordinates of its centroid.
Let p(1,−1), Q(2, −1), R(4,−3) be the mid-points of sides AB, BC and CA respectivelyof triangle ABC. Let A(x1, y1), B(x2, y3) be the vertices of triangle ABC. Then, p is the mid-point of AB∴ x1+x22=1, y1+y22=−1⇒ x1+x2=2 and y1+y2=−2 ...(i)Q is the mid-point of BC∴ x2+x32=2, y2+y32=−1⇒ x2+x3=4 and y2+y3=−2 ...(ii)R is the mid-point of AC∴ x1+x32=4, y1+y22=−3⇒ x1+x3=8 and y1+y3=−6 ...(iii)Adding equations (i), (ii) and (iii), we getx1+x2+x2+x3+x1+x3=2+4+8 and y1+y2+y2+y3+y1+y3=−2−2−6⇒ 2(x1+x2+x3)=14 and 2(y1+y2+y3)=−10⇒ x1+x2+x3=7 and y1+y2+y3=−5∴ Coordinates of centroid are (x1+x2+x33,y1+y2+y33)i.e. (73,−53)
Let p(1,−1), Q(2, −1), R(4,−3) be the mid-points of sides AB, BC and CA respectivelyof triangle ABC. Let A(x1, y1), B(x2, y3) be the vertices of triangle ABC. Then, p is the mid-point of AB∴ x1+x22=1, y1+y22=−1⇒ x1+x2=2 and y1+y2=−2 ...(i)Q is the mid-point of BC∴ x2+x32=2, y2+y32=−1⇒ x2+x3=4 and y2+y3=−2 ...(ii)R is the mid-point of AC∴ x1+x32=4, y1+y22=−3⇒ x1+x3=8 and y1+y3=−6 ...(iii)Adding equations (i), (ii) and (iii), we getx1+x2+x2+x3+x1+x3=2+4+8 and y1+y2+y2+y3+y1+y3=−2−2−6⇒ 2(x1+x2+x3)=14 and 2(y1+y2+y3)=−10⇒ x1+x2+x3=7 and y1+y2+y3=−5∴ Coordinates of centroid are (x1+x2+x33,y1+y2+y33)i.e. (73,−53)
