In the Cartesian coordinate plane shown, plot the points A(2,3), B(−2,−1), and C(2,−3), and connect them in order A→B→C. Then find the area of △ABC.
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Step-by-Step Solution
Step 1: Plot the points and draw the triangle
First, we plot the given points A(2,3), B(−2,−1), and C(2,−3) on the Cartesian coordinate plane. Then, we connect these points in the specified order to form triangle ABC.
Step 2: Identify the base and height of the triangle
Notice that points A(2,3) and C(2,−3) have the same x -coordinate. This means the side AC is a vertical line segment, which can serve as the base of the triangle. The length of the base AC is the absolute difference of their y -coordinates. The height of the triangle with respect to this base will be the perpendicular distance from point B to the line containing AC, which is the absolute difference of the x -coordinates of A (or C) and B.
Step 3: Calculate the length of the base
Using the coordinates of points A(2,3) and C(2,−3), we calculate the length of the base AC. The length is the absolute difference between their y -coordinates.
Step 4: Calculate the height of the triangle
The height of the triangle is the perpendicular distance from point B(−2,−1) to the line segment AC. Since AC lies on the line x=2, the height is the absolute difference between the x -coordinate of B and the x -coordinate of AC.
Step 5: Calculate the area of the triangle
Now we use the formula for the area of a triangle: Area=21×base×height. We substitute the calculated base length of 6 units and height of 4 units into the formula.
Step 6: Final Calculation
Performing the multiplication, we find the area of △ABC to be 12 square units.