If the points $A(x, y)$, $B(2, 3)$, and $C(3, 2)$ are vertices of a triangle whose area is $7\text{ sq. units}$, show that: $x+y=19\quad \text{or}\quad x+y=-9$

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 $A(x_1, y_1) = (x, y)$, $B(x_2, y_2) = (2, 3)$, and $C(x_3, y_3) = (3, 2)$ 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.

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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