If the points A(x, y), B(1,3), and C(3,1) are vertices of a triangle whose area is 8 sq. units, show that: x+y=12orx+y=−4
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Step-by-Step Solution
Step 1: Recall the Area of a Triangle Formula
The area of a triangle with vertices (x1,y1), (x2,y2), and (x3,y3) can be calculated using this formula. We will substitute the given coordinates and the area into this formula.
Step 2: Substitute Given Values into the Formula
We are given the vertices A(x, y), B(1,3), and C(3,1), and the area is 8 sq. units. We substitute these values into the area formula.
Step 3: Simplify the Expression
Multiply both sides by 2 to remove the fraction. Then, simplify the expression inside the absolute value by combining like terms.
Step 4: Solve for the two possible cases
Since the absolute value of an expression is 16, the expression itself can be either 16 or −16. This gives us two separate linear equations to solve.
Step 5: Solve the first equation
For the first case, add 8 to both sides and then divide the entire equation by 2 to simplify it.
Step 6: Solve the second equation
For the second case, add 8 to both sides and then divide the entire equation by 2 to simplify it.