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Question
A (6, 3), B (−3, 5), C (4, −2) and D (x, 3x) are four points. If ∆ DBC : ∆ ABC = 1 : 2, then x is equal to
Options
11/8
8/11
3
none of these
Solution
11/8
The area of a triangle with vertices D (x, 3x), B (−3, 5) and C (4, −2) is given below:
Area of ∆DBC = \[\frac{1}{2}\left\{ x\left( 5 + 2 \right) - 3\left( - 2 - 3x \right) + 4\left( 3x - 5 \right) \right\}\]
\[\Rightarrow\] Area of ∆DBC = \[\left( 14x - 7 \right) \text { sq units }\]
Similarly, the area of a triangle with vertices A (6, 3), B (−3, 5) and C (4, −2) is given below:
∆ABC = \[\frac{1}{2}\left\{ 6\left( 5 + 2 \right) - 3\left( - 2 - 3 \right) + 4\left( 3 - 5 \right) \right\}\]
\[\Rightarrow\] ∆ABC = \[\frac{49}{2}\] sq units
Given:
∆DBC:∆ABC = 1:2
\[\frac{2\left( 14x - 7 \right)}{49} = \frac{1}{2}\]
\[ \Rightarrow 8x - 4 = 7\]
\[ \Rightarrow x = \frac{11}{8}\]
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