If A=(2,3), B=(-2-5), C=(-4,6) and if P is a point on BC such that AP bisects the angle A, then find P.

Answer

To find the point P on line segment BC such that AP bisects angle A, we can follow these steps:

### Step 1: Identify the coordinates of points A, B, and C.
- A = (2, 3)
- B = (-2, -5)
- C = (-4, 6)

### Step 2: Calculate the lengths of sides AB and AC using the distance formula.
The distance formula between two points (x1, y1) and (x2, y2) is given by:
\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]

**Calculating AB:**
\[ AB = \sqrt{((-2) - 2)^2 + ((-5) - 3)^2} = \sqrt{(-4)^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5} \]

**Calculating AC:**
\[ AC = \sqrt{((-4) - 2)^2 + (6 - 3)^2} = \sqrt{(-6)^2 + (3)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \]

### Step 3: Find the ratio of the sides AB and AC.
The ratio \( \frac{AB}{AC} \) is:
\[ \frac{AB}{AC} = \frac{4\sqrt{5}}{3\sqrt{5}} = \frac{4}{3} \]

### Step 4: Use the section formula to find the coordinates of point P.
Since P divides BC in the ratio 4:3, we can use the section formula:
\[ P = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right) \]
where \( m = 4 \) and \( n = 3 \), \( B = (x_1, y_1) = (-2, -5) \), and \( C = (x_2, y_2) = (-4, 6) \).

**Calculating the x-coordinate of P:**
\[ P_x = \frac{4 \cdot (-4) + 3 \cdot (-2)}{4 + 3} = \frac{-16 - 6}{7} = \frac{-22}{7} \]

**Calculating the y-coordinate of P:**
\[ P_y = \frac{4 \cdot 6 + 3 \cdot (-5)}{4 + 3} = \frac{24 - 15}{7} = \frac{9}{7} \]

### Step 5: Combine the coordinates to get the point P.
Thus, the coordinates of point P are:
\[ P = \left( -\frac{22}{7}, \frac{9}{7} \right) \]

### Final Answer:
The coordinates of point P are \( P = \left( -\frac{22}{7}, \frac{9}{7} \right) \).

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