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Chapter 6: Problem 46

Find the area of the triangle with the given vertices. Round to the nearest square unit. $$(-2,-3),(-2,2),(2,1)$$

Short Answer

Expert verified
The area of the triangle is approximately 6 square units.

Step by step solution

01

Derive the Formula

The formula for the area of a triangle given its vertices at coordinates \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) is derived from the concept of determinant of a matrix. It is \[ Area = \frac{1}{2} | (x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)) |\]
02

Substitute the Coordinates

Substitute the coordinates of the given vertices into the formula. \((x_1, y_1) = (-2, -3)\), \((x_2, y_2) = (-2, 2)\), \((x_3, y_3) = (2, 1)\). Thus, the area becomes \[ Area = \frac{1}{2} | (-2 * (2 - 1) - 2 * (1 + 3) + 2*(-3 - 2)) |\]
03

Calculate the Area

After substituting the coordinates into the formula, simplify the equation to get the area of the triangle: \[ Area = \frac{1}{2} * 11 = 5.5 \space units\^2 \] Round up to the nearest unit results in 6 square units.

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