`A(1,3)and c(-2/5,-2/5)`are the vertices of a `DeltaABCand`the equation of the angle bisector of `/_ABC` is `x+y=2.`

Options

A. (A) (3/10, 17/10)
B. (B) (17/10, 3/10)
C. (C) (-5/2, 9/2)
D. (D) (-1,1)

Explanation

To solve the problem, we need to find the coordinates of point B given the vertices A(1, 3) and C(-2/5, -2/5) of triangle ABC, along with the equation of the angle bisector of angle ABC, which is x + y = 2.

### Step-by-Step Solution:

1. **Identify the Given Points:**
- Point A = (1, 3)
- Point C = (-2/5, -2/5)

2. **Equation of the Angle Bisector:**
- The angle bisector of angle ABC is given by the equation:
\[ x + y = 2 \]

3. **Finding the Slope of the Angle Bisector:**
- The slope of the line x + y = 2 can be rewritten in slope-intercept form (y = mx + b):
\[ y = -x + 2 \]
- Thus, the slope (m) of the angle bisector is -1.

4. **Finding the Slope of Line AC:**
- To find the slope of line AC, we can use the formula:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
- For points A(1, 3) and C(-2/5, -2/5):
\[ \text{slope of AC} = \frac{-2/5 - 3}{-2/5 - 1} = \frac{-2/5 - 15/5}{-2/5 - 5/5} = \frac{-17/5}{-7/5} = \frac{17}{7} \]

5. **Finding the Slope of Line AB:**
- Since the angle bisector divides the angle into two equal parts, we can use the relationship between the slopes of the lines:
\[ \text{slope of AB} = m_1 \text{ (slope of AC)} \text{ and } m_2 \text{ (slope of angle bisector)} \]
- Using the formula for the angle between two lines:
\[ \tan(\theta) = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right| \]
- Here, \( m_1 = \frac{17}{7} \) and \( m_2 = -1 \).

6. **Setting Up the Equation:**
- From the angle bisector property:
\[ \frac{m_1 - m_2}{1 + m_1 m_2} = \frac{\frac{17}{7} + 1}{1 - \frac{17}{7}} = \frac{\frac{24}{7}}{-\frac{10}{7}} = -\frac{24}{10} = -\frac{12}{5} \]

7. **Finding Coordinates of Point B:**
- Let the coordinates of point B be (h, k). The point B lies on the angle bisector line:
\[ h + k = 2 \]
- We also have the relationship derived from the angle bisector properties:
\[ \frac{h - 1}{k - 3} = -\frac{12}{5} \]

8. **Solving the System of Equations:**
- From \( h + k = 2 \), we can express k in terms of h:
\[ k = 2 - h \]
- Substitute k into the angle bisector equation:
\[ \frac{h - 1}{(2 - h) - 3} = -\frac{12}{5} \]
\[ \frac{h - 1}{-h - 1} = -\frac{12}{5} \]
- Cross-multiplying gives:
\[ 5(h - 1) = 12(-h - 1) \]
\[ 5h - 5 = -12h - 12 \]
\[ 17h = -7 \implies h = -\frac{7}{17} \]
- Substitute h back to find k:
\[ k = 2 - \left(-\frac{7}{17}\right) = 2 + \frac{7}{17} = \frac{34}{17} + \frac{7}{17} = \frac{41}{17} \]

9. **Final Coordinates of Point B:**
- Therefore, the coordinates of point B are:
\[ B\left(-\frac{7}{17}, \frac{41}{17}\right) \]