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Mathematics

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Answer

Area=24 square units\text{Area} = 24 \text{ square units}

Step-by-step solution

4 steps

  1. Identify the coordinates of the vertices.

    Let the vertices be A(x1,y1)=(1,−1)A(x_1, y_1) = (1, -1), B(x2,y2)=(−4,6)B(x_2, y_2) = (-4, 6), and C(x3,y3)=(−3,−5)C(x_3, y_3) = (-3, -5).

  2. Use the formula for the area of a triangle given its vertices.

    The area of a triangle with vertices (x1,y1)(x_1, y_1), (x2,y2)(x_2, y_2), and (x3,y3)(x_3, y_3) is given by:

    Area=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|

  3. Substitute the coordinates into the formula.

    Area=12∣1(6−(−5))+(−4)(−5−(−1))+(−3)(−1−6)∣\text{Area} = \frac{1}{2} |1(6 - (-5)) + (-4)(-5 - (-1)) + (-3)(-1 - 6)|

  4. Perform the calculations.

    First, calculate the terms inside the parentheses:

    (y2−y3)=6−(−5)=6+5=11(y_2 - y_3) = 6 - (-5) = 6 + 5 = 11 (y3−y1)=−5−(−1)=−5+1=−4(y_3 - y_1) = -5 - (-1) = -5 + 1 = -4 (y1−y2)=−1−6=−7(y_1 - y_2) = -1 - 6 = -7

    Now substitute these values back into the area formula:

    Area=12∣1(11)+(−4)(−4)+(−3)(−7)∣\text{Area} = \frac{1}{2} |1(11) + (-4)(-4) + (-3)(-7)| Area=12∣11+16+21∣\text{Area} = \frac{1}{2} |11 + 16 + 21| Area=12∣48∣\text{Area} = \frac{1}{2} |48| Area=12×48\text{Area} = \frac{1}{2} \times 48 Area=24\text{Area} = 24

    The area of the triangle is 24 square units.

    Area=24 square units\text{Area} = 24 \text{ square units}
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