Assertion- Area of the triangle whose vertices are $A − 3 2, 3, B (6, − 2)$ and $C (− 3, 4)$ andis 0.

Reason- The points A, B and $C$ are collinear.

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If both Assertion and Reason are correct and Reason is the correct explanation of Assertion.

If both Assertion and Reason are correct, but Reason is not the correct explanation of Assertion.

If Assertion is correct but Reason is incorrect.

If Assertion is incorrect but Reason is correct.

answer is A.

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Detailed Solution

We have been given,
Area of the triangle whose vertices are

A (\frac{- 3}{2}, 3), B (6, - 2)

and

C (- 3, 4)

is 0.
Points A, B and C are collinear.
Using the formula,

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

We have given,

A ≡ − 32, 3 B ≡ (6, − 2) C ≡ (− 3, 4)

So, the area of triangle ABC,

Here, x_{1} = (- \frac{3}{2}), y_{1} = 3, x_{2} = 6, y_{2} = - 2, x_{3} = (- 3), y_{3} = 4

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

\Rightarrow \frac{1}{2} |9 + 6 - 15|

\Rightarrow 0

Thus, the assertion is true.
Now,
Slope of AB =

\frac{- 2 - 3}{6 - (- \frac{3}{2})} = - \frac{2}{3}

Slope of BC =

\frac{4 - (- 2)}{- 3 - 6} = - \frac{2}{3}

As a result, the points A, B, and C are parallel.
Consequently, the reason is likewise true. Additionally, we are aware that the area of a triangle made up of three parallel points is always zero.
Therefore, assertion and reason are both correct and reason is the correct explanation of assertion.
Hence, option 1 is correct.

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