Area of a triangle in the plane: A=±(1)/(2)(x1 y2-x2 y1+x2 y3-x3 y2+x3 y1-x1 y3) Find the area o

step 1 This is our question number 47. This is the formula for area of triangle and the given three poi

Area of a triangle in the plane: A=±(1)/(2)(x1 y2-x2 y1+x2...

Area of a triangle in the plane:
$$A=\pm \frac{1}{2}\left(x_{1} y_{2}-x_{2} y_{1}+x_{2} y_{3}-x_{3} y_{2}+x_{3} y_{1}-x_{1} y_{3}\right)$$
Find the area of a triangle whose vertices are
$(6,-2),(-5,4),$ and (-1,7).

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Key Concepts

Coordinate Geometry

This concept involves representing geometric figures using ordered pairs or tuples in a coordinate system. It allows the use of algebraic methods to compute various properties of shapes, such as lengths, angles, and areas, by utilizing the coordinates of points that define the figure.

Triangle Area Calculation Using Coordinates

This technique computes the area of a triangle when the coordinates of its vertices are known. It involves substituting the coordinates into a formula that sums the products of the vertices in a cyclic manner, often resulting in an expression whose absolute value, when multiplied by one-half, gives the area of the triangle.

Shoelace Formula

The shoelace formula, also known as Gauss’s area formula, is a method for calculating the area of a polygon when the coordinates of its vertices are provided. For a triangle, it involves listing the vertices in order, multiplying in a crisscross pattern, summing these products, and then taking one-half of the absolute difference to obtain the area.

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