InterviewSolution
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InterviewSolution
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Four points A (6, 3), B (-3, 5), C (4, - 2) and D (x, 3x) are given in such a way that \(\frac{△DBC}{△ABC}\) = \(\frac{1}2\) , find x. |
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Answer» Four points A (6, 3), B (−3, 5) C (4, −2) and D(x, 3x) Area of the triangle having vertices (x1,y1), (x2,y2) and (x3,y3) = \(\frac{1}2\) |x1(y2-y3)+x2(y3-y1)+x3(y1-y2)| Area of ∆ABC = \(\frac{1}2\) |6(5 – (- 2)) - 3(-2 -3) + 4(3 – 5)| = \(\frac{1}2\) |42 + 15 -8| = \(\frac{49}2\) sq. units Area of ∆DBC = \(\frac{1}2\) |x(5 – (-2)) + 3(-2 -3x) + 4(3x - 5))| = \(\frac{1}2\) |7x +6 + 9x + 12x - 20| = \(\frac{1}2\) | 28x -14| = ± 7(2x -1 ) It is given that \(\frac{△DBC}{△ABC}\) = \(\frac{1}2\) ∴2 × ∆DBC = ∆ABC 2 × (± 7(2x -1 )) = \(\frac{49}2\) ∴ ± 4(2x -1 ) = 7 ∴ 4(2x -1 ) = 7 or -4(2x -1 ) = 7 ∴ 8x – 4 =7 or -8x + 4 =7 ∴ 8x =11 or -8x =3 ∴x = \(\frac{11}8\) or x = \(\frac{-3}8\) Hence, the value of x is \(\frac{11}8\) or \(\frac{-3}8\) |
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