If the points A(x, y), B(2,3), and C(3,2) are vertices of a triangle whose area is 7 sq. units, show that: x+y=19orx+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 its vertices, we use the determinant formula. The absolute value ensures the area is always positive. We are given the vertices A(x, y), B(2,3), and C(3,2), and the area is 7 square units.
Step 2: Substitute the Given Coordinates and Area
Now, we substitute the coordinates of points A(x1,y1)=(x,y), B(x2,y2)=(2,3), and C(x3,y3)=(3,2) into the area formula. We also set the area equal to 7.
Step 3: Simplify the Expression Inside the Absolute Value
First, multiply both sides by 2 to remove the fraction. Then, simplify the terms inside the absolute value by performing the multiplications and combining like terms.
Step 4: Solve for Two Possible Cases
Since the expression is inside an absolute value, it can be either positive or negative 14. This gives us two separate linear equations to solve.
Step 5: Solve Each Equation for x + y
For the first case, add 5 to both sides of the equation x+y−5=14. For the second case, add 5 to both sides of the equation x+y−5=−14. This yields the two required relationships.