Verify that area of the triangle with vertices (4,6) (7,10) and(1,-2) remains invariant under the translation of axes when the origin is shifted to the point (-2,1).

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Solution

$Let the co-ordinate of the vertex be A(4,6) B(7,10)and C(1,-2) N o w a r e a o f t h e Δ A B C i s g i v e n b y Δ = \frac{1}{2} | (x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}))) | = \frac{1}{2} | (4 (10 + 2) + 7 (- 2 - 6) + 1 (6 - 10)) | = \frac{1}{2} | (48 - 56 - 4) | = 6 After transforming the origin to(-2,1) the co-ordinate of the vertex wil be A (2, 7), (5, 11) a n d C (- 1, - 1) . N o w t h e a r e a w i l l b e Δ = \frac{1}{2} | (x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}))) | = \frac{1}{2} | (2 (11 + 1) + 5 (- 1 - 7) - 1 (+ 7 - 11)) | = \frac{1}{2} | (24 - 40 + 4) | = 6 H e r e Δ = Δ, H e n c e p r o v e d .$

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