Last updated at Jan. 31, 2020 by
Example 13 The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are (3, –5, 7) and (–1, 7, –6), respectively, find the coordinates of the point C. Let ABC be a triangle where A (3, –5, 7) , B( –1, 7, –6) Let G be the centroid of ∆ ABC So, G (1, 1, 1) Let Coordinate of C (x, y, z) We know that Co ordinate of centroid whose vertices are (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) is ((𝑥_1 + 𝑦_1 + 𝑧_1)/3,(𝑥_2 + 𝑦_2 + 𝑧_2)/3,(𝑥_3 + 𝑦_3 + 𝑧_3)/3) Here, x1 = 3 , y1 = –5 , z1 = 7 x2 = –1 , y2 = 7 , z2 = –6 x3 = x , y2 = y , z3 = z ∴ Coordinates of G (1, 1, 1) = ((3 + (−1) + 𝑥)/3,(−5 + 7 + 𝑦)/3,(7 + (−6) + 𝑧)/3) (1, 1, 1) = ((2 + 𝑥)/3,(2 + 𝑦)/3,(1 + 𝑧)/3) x – coordinate 1 = (2 + 𝑥)/3 3(1) = 2 + x x = 1 y – coordinate 1 = (2 + 𝑦)/3 3(1) = 2 + y y = 1 z – coordinate 1 = (1 + 𝑧)/3 3(1) = 1 + z y = 2 Hence Required coordinate of C = (x, y, z) = (1, 1, 2)
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Two vertices of ABC are A 1,4 and B 5,2 and its centroid is G 0, 3. Then, the coordinates of C area 4,3b 4,15c 4, 15d 15, 4
Solution
Two vertices of ΔABCare A(-1,4) and B(5,2). Let the third vertex be C (a, b)
Then the co-ordinates of its centroid are
C= (−1+5+a)3,(4+2+b) 3
C= 4+a3,6+b3
But it is given that G(0,-3) is the centroid. Therefore
0=4+a3,−3=6+b3
⇒a=-4 , -9-6=b
⇒⇒ a = -4, b = -15
Therefore, the third vertex of ΔABCis C(-4,-15)
Mathematics
Secondary School Mathematics X
Standard X