If two vertices of a triangle are (1,3) and (4,-1) and the area of triangle is 5 sq. units, then the angle at the third vertex lies in :

Options

A. `(0,tan^-1.5/4]`
B. `(0,tan^-1.5/4)`
C. `(2tan^-1"5/4,2)`
D. none of these

Explanation

To solve the problem step by step, we will find the angle at the third vertex of the triangle given the coordinates of two vertices and the area of the triangle.

### Step 1: Identify the vertices
Let the vertices of the triangle be:
- Vertex A: \( (1, 3) \)
- Vertex B: \( (4, -1) \)
- Vertex C: \( (x, y) \) (coordinates of the third vertex which we need to determine)

### Step 2: Calculate the length of side BC
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
We calculate the length of side BC:
\[ BC = \sqrt{(4 - 1)^2 + (-1 - 3)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Use the area formula to find the altitude
The area \( A \) of a triangle can also be expressed as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Given that the area is 5 square units and the base \( BC = 5 \):
\[ 5 = \frac{1}{2} \times 5 \times h \]
Solving for height \( h \):
\[ 5 = \frac{5h}{2} \implies 10 = 5h \implies h = 2 \]

### Step 4: Determine the coordinates of the midpoint M of BC
The midpoint \( M \) of line segment \( BC \) can be calculated as:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{1 + 4}{2}, \frac{3 + (-1)}{2} \right) = \left( \frac{5}{2}, 1 \right) \]

### Step 5: Set up the triangle and find the angles
We drop a perpendicular from point A to line BC, meeting at point M. The altitude \( AM \) is 2 units long.

### Step 6: Use trigonometric ratios to find angle A
In triangle \( AMB \):
- The opposite side (BM) is half the length of BC, which is \( \frac{5}{2} \).
- The adjacent side (AM) is the altitude, which is 2.

Using the tangent function:
\[ \tan\left(\frac{A}{2}\right) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BM}{AM} = \frac{\frac{5}{2}}{2} = \frac{5}{4} \]

### Step 7: Solve for angle A
To find angle \( A \):
\[ \frac{A}{2} = \tan^{-1}\left(\frac{5}{4} \right) \]
Thus,
\[ A = 2 \tan^{-1}\left(\frac{5}{4}\right) \]

### Conclusion
The angle at the third vertex lies in the range determined by \( 2 \tan^{-1}\left(\frac{5}{4}\right) \).