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If the origin is the centroid of a triangle ABC having vertices A(a, 1, 3), B(-2, b, -5) and C(4, 7, c), find the values of a, b, c. |
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Answer» Given: The coordinates of the A, B and C of the triangle ABC are (a, 1, 3), (-2, b, -5) and (4, 7, c) respectively. The centroid of the triangle is (0, 0, 0) To find: the values of a, b, c Formula used: Centroid of triangle ABC whose vertices are A(x1, y1, z1), B(x2, y2, z2) and C(x3, y3, z3) is given by, \(\Big(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_2}{3}\Big)\) Here A(a, 1, 3), B(-2, b, -5) and C(4, 7, c) Centroid of the triangle (0,0,0) = \(\Big(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_2}{3}\Big)\) ⇒ (0,0,0) = \(\Big(\frac{a-2+4}{3},\frac{-2+b+7}{3},\frac{3-5+c}{3}\Big)\) ⇒ (0,0,0) = \(\Big(\frac{a+2}{3},\frac{b+5}{3},\frac{c-2}{3}\Big)\) On comparing: \(\frac{a+2}{3}\) = 0 ⇒ a + 2 = 0 ⇒ a = –2 \(\frac{b+5}{3}\) = 0 ⇒ b + 5 = 0 ⇒ b = –5 \(\frac{c-2}{3}\) = 0 ⇒ c – 2 = 0 ⇒ c = 2 Hence, values of a, b and c are -2, -5, 2 |
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