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State whether the given statement is true or false.

If $A (− 5, 7), B (− 4, − 5), C (− 1, − 6)$ and $D (4, 5)$ andare the vertices of a quadrilateral, then the area of the quadrilateral ABCD is 0.

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True

False

answer is B.

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Detailed Solution

We have been given four points A(-5,7), B(-4, -5), C(-1, -6) and D(4, 5).
As a result, if the quadrilateral is divided into two triangles, we get

Δ A B C, Δ A C D

.
Now, 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} = (- 5), y_{1} = 7, x_{2} = (- 4), y_{2} = - 5, x_{3} = (- 1), y_{3} = (- 6)

, in the formula,

ar (ΔABC) = \frac{1}{2} |(- 5) (- 5 - (- 6)) + (- 4) (- 6 - (7)) + (- 1) (7 - (- 5))|

ar (ΔABC) = \frac{1}{2} |- 5 + 52 - 12|

ar (ΔABC) = \frac{35}{2}

Calculating the area of the triangle ACD.

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

Substituting

x_{1} = (- 5), y_{1} = 7, x_{2} = (- 1), y_{2} = - 6, x_{3} = 4, y_{3} = 5

, in the formula,

ar (ΔACD) = \frac{1}{2} |(- 5) (- 6 - 5) + (- 1) (5 - 7) + 4 (7 - (- 6))|

ar (ΔACD) = \frac{1}{2} |55 + 2 + 52|

ar (ΔACD) = \frac{109}{2}

Calculating the area of the quadrilateral,

ar (ΔABC) + ar (ΔACD) = \frac{35}{2} + \frac{109}{2}

ar (ΔABC) + ar (ΔACD) = \frac{144}{2}

ar (ΔABC) + ar (ΔACD) = 72

Therefore, the given statement is false.
The area of the quadrilateral ABCD is 72 units.
Hence, option 2 is correct.

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