If the points $A(x, y)$, $B(1, 3)$, and $C(3, 1)$ are vertices of a triangle whose area is $8\text{ sq. units}$, show that: $x+y=12\quad \text{or}\quad x+y=-4$

Step 1: Recall the Area of a Triangle Formula

The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ 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.

More from this chapter

• Q7. If A(a, 0), B(1, 1) and C(0, b) form a triangle, right angled at B when joined, then establish a relation between a and b. [2 Marks]
• Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by the point (-1, 6).
• Find a point on the y-axis that is at the same distance from the points (-5, 2) and (9, -2).
• Find the coordinates of the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1 internally.
• Find the relation between x and y if the point (x, y) is equidistant from (7, 1) and (3, 5).
• 1. Find the distance between the following pairs of points: (i) (2, 3), (4, 1) (ii) (-5, 7), (-1, 3) (iii) (a, b), (-a, -b)
• Find the coordinates of the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4).
• Find the ratio in which the line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis, and find the coordinates of the point of division.

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