If the points A(x, y), B(1,2), and C(2,1) are vertices of a triangle whose area is 6 sq. units, show that: x+y=15orx+y=−9
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Step-by-Step Solution
Step 1: Recall the Area of a Triangle Formula
To find the area of a triangle given the coordinates of its vertices, we use a specific formula. This formula involves the coordinates of the three vertices (x1,y1), (x2,y2), and (x3,y3). The absolute value ensures that the area is always positive.
Step 2: Substitute the Given Coordinates and Area
We are given the vertices A(x, y), B(1,2), and C(2,1), and the area is 6 square units. We substitute these values into the area formula. Here, (x1,y1)=(x,y), (x2,y2)=(1,2), and (x3,y3)=(2,1).
Step 3: Simplify the Expression
First, we multiply both sides by 2 to remove the fraction. Then, we simplify the expression inside the absolute value by performing the multiplications and combining like terms. This gives us a simpler equation involving x and y.
Step 4: Solve for Two Possible Cases
Since the absolute value of an expression is 12, the expression itself can be either 12 or −12. This leads to two separate linear equations that we need to solve.
Step 5: Find the Final Equations
By isolating x+y in each of the two equations, we arrive at the two possible relationships between x and y. These are the required conditions for the area of the triangle to be 6 square units.