Let A(4,−2), B(1,1) and C(9,−3) be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E, and F on the sides BC, CA and AB of the triangle ABC respectively, is:
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Step-by-Step Solution
Step 1: Calculate the area of triangle ABC
To find the area of the parallelogram, we first need to calculate the area of the triangle ABC. We use the determinant formula for the area of a triangle given its vertices (x1,y1), (x2,y2), and (x3,y3).
Step 2: Substitute coordinates and compute area
Substitute the coordinates A(4,−2), B(1,1), and C(9,−3) into the formula. After performing the arithmetic operations, we find the area of triangle ABC to be 6 square units.
Step 3: Relate parallelogram area to triangle area
For a parallelogram AFDE inscribed in a triangle ABC with vertices D, E, and F on sides BC, CA, and AB respectively, the maximum area of the parallelogram is half the area of the triangle. This is a known geometric property.
Step 4: Calculate maximum area of parallelogram
Using the property that the maximum area of the inscribed parallelogram is half the area of the triangle, we multiply the calculated area of triangle ABC by 21 to find the maximum area of parallelogram AFDE.