If the points A(x, y), B(2,4), and C(4,2) are vertices of a triangle whose area is 9 sq. units, show that: x+y=15orx+y=−3
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Step-by-Step Solution
Step 1: Recall the Area of a Triangle Formula
To find the area of a triangle with given vertices, we use the determinant formula. The absolute value ensures the area is always positive. We are given the vertices A(x, y), B(2,4), and C(4,2), and the area is 9 square units.
Step 2: Substitute the Given Values into the Formula
Now, we substitute the coordinates of the points A(x1,y1)=(x,y), B(x2,y2)=(2,4), and C(x3,y3)=(4,2) into the area formula. We also set the area equal to 9.
Step 3: Simplify the Expression
We multiply both sides by 2 to remove the fraction and then simplify the expression inside the absolute value by combining like terms. This gives us a simplified equation involving x and y.
Step 4: Solve for Two Possible Cases
Since the absolute value of an expression is 18, the expression itself can be either 18 or −18. This leads to two separate linear equations.
Step 5: Solve the First Case
For the first case, we add 12 to both sides and then divide the entire equation by 2 to get the first relationship between x and y.
Step 6: Solve the Second Case
For the second case, we again add 12 to both sides and then divide by 2 to obtain the second relationship between x and y. This completes the proof.