To find the equation of the angle bisector of angle A in triangle ABC with vertices A(-2, 4), B(5, 5), and C(4, -2), we will follow these steps:
### Step 1: Identify the coordinates of the vertices
The vertices of triangle ABC are given as:
- A(-2, 4)
- B(5, 5)
- C(4, -2)
### Step 2: Calculate the lengths of sides AB and AC
We will use the distance formula to find the lengths of sides AB and AC.
**Distance Formula:**
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
**Length of AB:**
- \( A(-2, 4) \) and \( B(5, 5) \)
\[ AB = \sqrt{(5 - (-2))^2 + (5 - 4)^2} \]
\[ = \sqrt{(5 + 2)^2 + (5 - 4)^2} \]
\[ = \sqrt{7^2 + 1^2} \]
\[ = \sqrt{49 + 1} \]
\[ = \sqrt{50} \]
\[ = 5\sqrt{2} \]
**Length of AC:**
- \( A(-2, 4) \) and \( C(4, -2) \)
\[ AC = \sqrt{(4 - (-2))^2 + (-2 - 4)^2} \]
\[ = \sqrt{(4 + 2)^2 + (-2 - 4)^2} \]
\[ = \sqrt{6^2 + (-6)^2} \]
\[ = \sqrt{36 + 36} \]
\[ = \sqrt{72} \]
\[ = 6\sqrt{2} \]
### Step 3: Use the Angle Bisector Theorem
According to the Angle Bisector Theorem:
\[ \frac{AB}{AC} = \frac{BD}{DC} \]
Where \( D \) is the point on side \( BC \) that divides it in the ratio of the lengths of \( AB \) and \( AC \):
\[ \frac{AB}{AC} = \frac{5\sqrt{2}}{6\sqrt{2}} = \frac{5}{6} \]
### Step 4: Find the coordinates of point D using the section formula
Using the section formula, the coordinates of point D dividing BC in the ratio \( 5:6 \) can be calculated as follows:
\[ D\left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}\right) \]
Where \( m = 5 \) and \( n = 6 \).
Coordinates of B are \( (5, 5) \) and C are \( (4, -2) \):
\[ D\left(\frac{5 \cdot 4 + 6 \cdot 5}{5 + 6}, \frac{5 \cdot (-2) + 6 \cdot 5}{5 + 6}\right) \]
\[ = D\left(\frac{20 + 30}{11}, \frac{-10 + 30}{11}\right) \]
\[ = D\left(\frac{50}{11}, \frac{20}{11}\right) \]
### Step 5: Find the equation of line AD
Now, we will find the equation of the line passing through points A(-2, 4) and D\(\left(\frac{50}{11}, \frac{20}{11}\right)\).
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Where \( m \) is the slope given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Calculating the slope \( m \):
\[ m = \frac{\frac{20}{11} - 4}{\frac{50}{11} - (-2)} \]
\[ = \frac{\frac{20}{11} - \frac{44}{11}}{\frac{50}{11} + \frac{22}{11}} \]
\[ = \frac{-\frac{24}{11}}{\frac{72}{11}} \]
\[ = -\frac{24}{72} = -\frac{1}{3} \]
Now substituting into the point-slope form:
\[ y - 4 = -\frac{1}{3}(x + 2) \]
### Step 6: Rearranging to standard form
Multiply through by 3 to eliminate the fraction:
\[ 3(y - 4) = -1(x + 2) \]
\[ 3y - 12 = -x - 2 \]
\[ x + 3y - 10 = 0 \]
### Final Answer
The equation of the bisector of angle A is:
\[ x + 3y - 10 = 0 \]
