The area of a triangle is 5. Two of its vertices are (2, 1) and (3, - 2). The third vertex is (x, y) where y = x + 3. Then the co-ordinate of the third vertex is

Options

A. `((7)/(2),(13)/(2))or(-(3)/(2),(3)/(2))`
B. `(-(7)/(2),(13)/(2))or((3)/(2),(3)/(2))`
C. `((1)/(2),(3)/(2))`
D. `((3)/(2),(1)/(2))`

Answer

To find the coordinates of the third vertex of the triangle, we can follow these steps:

### Step 1: Identify the vertices and the area formula
We have two vertices of the triangle:
- A(2, 1)
- B(3, -2)
- C(x, y) where y = x + 3.

The area of a triangle given vertices (x1, y1), (x2, y2), (x3, y3) is given by the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

### Step 2: Substitute the known values into the area formula
Substituting the coordinates of A and B into the area formula:
\[ \text{Area} = \frac{1}{2} \left| 2(-2 - y) + 3(y - 1) + x(1 - (-2)) \right| \]
This simplifies to:
\[ \text{Area} = \frac{1}{2} \left| 2(-2 - y) + 3(y - 1) + 3x \right| \]

### Step 3: Set the area equal to 5
Given that the area of the triangle is 5, we have:
\[ 5 = \frac{1}{2} \left| 2(-2 - y) + 3(y - 1) + 3x \right| \]
Multiplying both sides by 2 gives:
\[ 10 = \left| 2(-2 - y) + 3(y - 1) + 3x \right| \]

### Step 4: Remove the absolute value
This leads to two equations:
1. \( 2(-2 - y) + 3(y - 1) + 3x = 10 \)
2. \( 2(-2 - y) + 3(y - 1) + 3x = -10 \)

### Step 5: Simplify the first equation
Let's simplify the first equation:
\[ 2(-2 - y) + 3(y - 1) + 3x = 10 \]
Expanding gives:
\[ -4 - 2y + 3y - 3 + 3x = 10 \]
Combining like terms:
\[ 3x + y - 7 = 10 \implies 3x + y = 17 \quad \text{(Equation 1)} \]

### Step 6: Simplify the second equation
Now simplifying the second equation:
\[ 2(-2 - y) + 3(y - 1) + 3x = -10 \]
Expanding gives:
\[ -4 - 2y + 3y - 3 + 3x = -10 \]
Combining like terms:
\[ 3x + y - 7 = -10 \implies 3x + y = -3 \quad \text{(Equation 2)} \]

### Step 7: Substitute y = x + 3 into the equations
Since \( y = x + 3 \), we substitute this into both equations.

For Equation 1:
\[ 3x + (x + 3) = 17 \implies 4x + 3 = 17 \implies 4x = 14 \implies x = \frac{7}{2} \]
Then,
\[ y = \frac{7}{2} + 3 = \frac{7}{2} + \frac{6}{2} = \frac{13}{2} \]

For Equation 2:
\[ 3x + (x + 3) = -3 \implies 4x + 3 = -3 \implies 4x = -6 \implies x = -\frac{3}{2} \]
Then,
\[ y = -\frac{3}{2} + 3 = -\frac{3}{2} + \frac{6}{2} = \frac{3}{2} \]

### Step 8: Final coordinates of the third vertex
Thus, the coordinates of the third vertex C can be:
1. \( C\left(\frac{7}{2}, \frac{13}{2}\right) \)
2. \( C\left(-\frac{3}{2}, \frac{3}{2}\right) \)

### Conclusion
The coordinates of the third vertex are either \( \left(\frac{7}{2}, \frac{13}{2}\right) \) or \( \left(-\frac{3}{2}, \frac{3}{2}\right) \).

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