In Exercises 45–46, find the area of the triangle with the given vertices. Round to the nearest square unit. (-2, -3), (-2, 2), (2, 1)

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Identify the coordinates of the vertices of the triangle: \(A(-2, -3)\), \(B(-2, 2)\), and \(C(2, 1)\).

Use the formula for the area of a triangle given coordinates \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\): \(\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|\)

Substitute the coordinates into the formula: \(\text{Area} = \frac{1}{2} \left| (-2)(2 - 1) + (-2)(1 - (-3)) + 2((-3) - 2) \right|\)

Simplify the expression inside the absolute value by performing the operations inside the parentheses and then the multiplications.

Calculate the absolute value of the simplified expression, multiply by \(\frac{1}{2}\), and then round the result to the nearest whole number to find the area of the triangle.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Coordinate Geometry and Triangles

Understanding how to represent points in the coordinate plane is essential. Each vertex of the triangle is given as an (x, y) coordinate, which allows the use of algebraic methods to calculate distances, slopes, and areas.

Area of a Triangle Using Coordinates

The area of a triangle with vertices at coordinates can be found using the formula: Area = 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. This formula uses the determinant method to calculate the absolute value of the signed area.

Rounding and Approximation

After calculating the exact area, rounding to the nearest square unit is necessary. This involves understanding decimal values and applying standard rounding rules to present the final answer in a simplified, practical form.