The position vectors of vertices of `triangle ABC` are `(1, -2), (-7,6) and (11/5, 2/5)` respectively. The measure of the interior angle A of the `triangle ABC,` is,

Options

A. acute and lies in `(75^(@), 90^(@))`
B. acute and lies in `(60^(@), 75^(@))`
C. acute and lies in `(45^(@), 60^(@))`
D. obtuse and lies in `(120^(@), 150^(@))`

Explanation

To find the measure of the interior angle A of triangle ABC with given position vectors, we can follow these steps:

### Step 1: Identify the vertices
The vertices of triangle ABC are given as:
- A(1, -2)
- B(-7, 6)
- C(11/5, 2/5)

### Step 2: Calculate the slopes of sides AB and AC
To find the angle A, we need to calculate the slopes of sides AB and AC.

**Slope of AB (M1)**:
Using the formula for slope \( M = \frac{y_2 - y_1}{x_2 - x_1} \):
- \( A(1, -2) \) and \( B(-7, 6) \)
- \( M1 = \frac{6 - (-2)}{-7 - 1} = \frac{6 + 2}{-8} = \frac{8}{-8} = -1 \)

**Slope of AC (M2)**:
Using the same formula:
- \( A(1, -2) \) and \( C(11/5, 2/5) \)
- \( M2 = \frac{2/5 - (-2)}{11/5 - 1} = \frac{2/5 + 10/5}{11/5 - 5/5} = \frac{12/5}{6/5} = 2 \)

### Step 3: Use the slopes to find the angle A
The angle A can be found using the formula:
\[ \tan A = \frac{M1 - M2}{1 + M1 \cdot M2} \]
Substituting the values:
\[ \tan A = \frac{-1 - 2}{1 + (-1) \cdot 2} = \frac{-3}{1 - 2} = \frac{-3}{-1} = 3 \]

### Step 4: Calculate the angle A
To find the angle A, we take the arctangent:
\[ A = \tan^{-1}(3) \]

### Step 5: Approximate the angle
Using a calculator or trigonometric tables:
\[ A \approx 72^\circ \]

### Conclusion
The measure of the interior angle A of triangle ABC is approximately \( 72^\circ \).

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