Question

use the following linear equations. line a: \(y=-x+2\) line \(b: y=-1\) line \(c: y=x+2\) Graph each equation in the same coordinate plane. What type of polygon do they form? Find the area of the polygon. Justify your method.

Short answer

The three lines form a triangle. After calculating the vertices of the triangle, we can find the area using the formula for the area of a triangle.

Step-by-step solution

Graphing the lines

First, plot the three lines on the same coordinate plane. The equation of line a is \(y=-x+2\), line b is \(y=-1\), and line c is \(y=x+2\). Each of these lines corresponds to a different slope and y-intercept. Plot each line accordingly.

Identify the Polygon

Observe where the three lines intersect. The points of intersection form the vertices of a polygon. In this case, the lines intersect at three points, so they form a triangle.

Calculate the coordinates of the vertices

The vertices of the triangle are the points of intersection between the lines. Solve the systems of linear equations two at a time to find these vertices. These are namely, a and b, b and c, a and c.

Calculate the Area of the Polygon

Once the vertices are found, apply the formula for the area of a triangle which is 0.5 multiplied by the absolute of the determinant of a \(2 \times 3\) matrix, with each row in the matrix represents the coordinates of a vertex of the triangle in homogeneous coordinates.