If the co-ordinates of the vertices A,B,C of the triangle ABC are `(-4, 2) , (12,-2) and (8,6)` respectively, then`angle B=`

Options

A. `tan ^(-1)""(-(6)/7)`
B. `tan ^(-1)""((6)/(7))`
C. `tan ^(-1)""(-(7)/(6))`
D. ` tan ^(-1) ""((7)/(6))`

Answer

To find the angle \( B \) in triangle \( ABC \) with vertices \( A(-4, 2) \), \( B(12, -2) \), and \( C(8, 6) \), we can use the slopes of the lines \( AB \) and \( BC \). The angle \( B \) can be calculated using the formula for the tangent of the angle between two lines.

### Step 1: Calculate the slope of line \( AB \)

The slope \( m_{AB} \) between points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:

\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the coordinates of points \( A(-4, 2) \) and \( B(12, -2) \):

\[ m_{AB} = \frac{-2 - 2}{12 - (-4)} = \frac{-4}{12 + 4} = \frac{-4}{16} = -\frac{1}{4} \]

### Step 2: Calculate the slope of line \( BC \)

Now, we calculate the slope \( m_{BC} \) between points \( B(12, -2) \) and \( C(8, 6) \):

\[ m_{BC} = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the coordinates of points \( B \) and \( C \):

\[ m_{BC} = \frac{6 - (-2)}{8 - 12} = \frac{6 + 2}{8 - 12} = \frac{8}{-4} = -2 \]

### Step 3: Use the tangent formula to find angle \( B \)

The formula for the tangent of the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is:

\[ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]

Substituting \( m_{AB} = -\frac{1}{4} \) and \( m_{BC} = -2 \):

\[ \tan B = \left| \frac{-2 - \left(-\frac{1}{4}\right)}{1 + \left(-\frac{1}{4}\right)(-2)} \right| \]

Calculating the numerator:

\[ -2 + \frac{1}{4} = -\frac{8}{4} + \frac{1}{4} = -\frac{7}{4} \]

Calculating the denominator:

\[ 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} \]

Now substituting back into the formula:

\[ \tan B = \left| \frac{-\frac{7}{4}}{\frac{3}{2}} \right| = \left| -\frac{7}{4} \cdot \frac{2}{3} \right| = \frac{7 \cdot 2}{4 \cdot 3} = \frac{14}{12} = \frac{7}{6} \]

### Step 4: Find angle \( B \)

To find angle \( B \), we take the arctangent:

\[ B = \tan^{-1}\left(\frac{7}{6}\right) \]

### Final Answer

Thus, the angle \( B \) is:

\[ B = \tan^{-1}\left(\frac{7}{6}\right) \]