To find the coordinates of point D, where AD is the angle bisector of angle BAC in triangle ABC with vertices A(2,1), B(6,-2), and C(8,9), we can follow these steps:
### Step 1: Calculate the lengths of sides AB and AC
We will use the distance formula to find the lengths of sides AB and AC.
The distance formula is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
**Calculating AB:**
- Coordinates of A: (2, 1)
- Coordinates of B: (6, -2)
\[
AB = \sqrt{(6 - 2)^2 + (-2 - 1)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
**Calculating AC:**
- Coordinates of A: (2, 1)
- Coordinates of C: (8, 9)
\[
AC = \sqrt{(8 - 2)^2 + (9 - 1)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
### Step 2: Apply the Angle Bisector Theorem
According to the Angle Bisector Theorem, the ratio of the lengths of the two segments created by the angle bisector is equal to the ratio of the lengths of the other two sides.
Let \( BD = m \) and \( DC = n \). Then:
\[
\frac{AB}{AC} = \frac{m}{n} = \frac{5}{10} = \frac{1}{2}
\]
This means \( m:n = 1:2 \).
### Step 3: Use the Section Formula
Using the section formula to find the coordinates of point D, which divides BC in the ratio \( m:n = 1:2 \).
The section formula states that if a point D divides the line segment joining points B(x1, y1) and C(x2, y2) in the ratio m:n, then the coordinates of D are given by:
\[
D\left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}\right)
\]
**Coordinates of B:** (6, -2)
**Coordinates of C:** (8, 9)
**m = 1, n = 2**
Calculating the coordinates of D:
\[
D_x = \frac{1 \cdot 8 + 2 \cdot 6}{1 + 2} = \frac{8 + 12}{3} = \frac{20}{3}
\]
\[
D_y = \frac{1 \cdot 9 + 2 \cdot (-2)}{1 + 2} = \frac{9 - 4}{3} = \frac{5}{3}
\]
Thus, the coordinates of point D are:
\[
D\left(\frac{20}{3}, \frac{5}{3}\right)
\]
### Final Answer
The coordinates of D are \( \left(\frac{20}{3}, \frac{5}{3}\right) \).
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