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If four points $A (6, 3), B (− 3, 5), C (4, − 2)$ and $D (x, 3 x)$ and form a triangle such that, $Δ D B C Δ A B C = 12$ form a triangle such that, ,then the value of x is ____.

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Detailed Solution

If four points

A (6, 3), B (− 3, 5), C (4, − 2)

and

D (x, 3 x)

form a triangle such that,

Δ D B C Δ A B C = 12

,then the value of x is

\frac{11}{8}

.
We have been given,
The four points

A (6, 3), B (− 3, 5), C (4, − 2)

and

D (x, 3 x)

forms a triangle such that

Δ D B C Δ A B C = 12

.
Calculating the area of triangle DBC.

ar (ΔDBC) = \frac{1}{2} [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})]

Substituting

x_{1} = x, x_{2} = (- 3), x_{3} = 4, y_{1} = 3 x, y_{2} = 5, and y_{3} = - 2

, in the formula,

ar (ΔDBC) = \frac{1}{2} |x (5 - (- 2)) + (- 3) (- 2 - 3 x) + 4 (3 x - 5)|

ar (ΔDBC) = \frac{1}{2} | 7 x + 6 + 9 x + 12 x - 20 |

ar (ΔDBC) = \frac{1}{2} | 28 x - 14 |

ar (ΔDBC) = | 14 x - 7 |

Similarly, calculating the area of triangle ABC,

ar (ΔABC) = \frac{1}{2} [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})]

Substituting

x_{1} = 6, x_{2} = - 3, x_{3} = 4, y_{1} = 3, y_{2} = 5, and y_{3} = - 2

, in the formula,

ar (ΔABC) = \frac{1}{2} |6 (5 - (- 2)) + (- 3) (- 2 - 3) + 4 (3 - 5)|

ar (ΔABC) = \frac{1}{2} | 42 + 15 - 8 |

ar (ΔABC) = \frac{1}{2} | 49 |

ar (ΔABC) = \frac{49}{2}

But we have been given that,

Δ D B C Δ A B C = 12

Therefore,
Question Image

As the value of x must be positive, therefore, the value of ‘x’ is

\frac{11}{8}

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