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What is the area of the triangle ABC?

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answer is D.

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Detailed Solution

We have been given,
Triangle ABC.
Question Image

Using the distance formula

D = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}

.
A scalene triangle has a different length of each side.
As we know, the vertices of the triangle are,

A 1, − 1, B − 4, 6, C − 3, − 5

Now, distance of AB, BC and AC,
For AB substituting

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

, in the formula,

A B = (− 4 − 1) 2 + (6 − (− 1)) 2 = 25 + 49 = 74

For BC substituting

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

, in the formula,

BC = \sqrt{{((- 3) - (- 4))}^{2} + {(- 5 - (- 1))}^{2}} = \sqrt{1 + 121} = \sqrt{122}

For AC substituting

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

, in the formula,

AC = \sqrt{{((- 3) - 1)}^{2} + {(- 5 - (- 1))}^{2}} = \sqrt{32} = 4 \sqrt{2}

Hence,

A B ≠ B C ≠ A C

So,

Δ A B C

is a scalene triangle.
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})]

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

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

\Rightarrow \frac{1}{2} |11 + 16 + 21|

\Rightarrow 24

∴ Δ A B C

is a scalene triangle and

Δ A B C = 24

square unit.
Hence, option 4 is correct.

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