Let in `DeltaABC` the coordinates of A are (0, 0). Internal angle bisector of `angle ABC` is `x-y+1=0` and mid - point of BC is `(-1, 3)`. Then, the ordinate of C is

Answer

To find the ordinate of point C in triangle ABC, we can follow these steps:

### Step 1: Understand the Given Information
We know:
- The coordinates of point A are (0, 0).
- The internal angle bisector of angle ABC is given by the equation \(x - y + 1 = 0\), which can be rewritten as \(y = x + 1\).
- The midpoint of segment BC is given as (-1, 3).

### Step 2: Determine the Coordinates of Point B
Since the angle bisector divides the angle at B into two equal angles and is at a 45-degree angle (slope = 1), we can assume that point B lies on the x-axis. Thus, the coordinates of point B can be represented as \((-1, 0)\).

### Step 3: Use the Midpoint Formula
Let the coordinates of point C be \((h, k)\). The midpoint M of segment BC can be calculated using the midpoint formula:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Substituting the coordinates of B and C:
\[ (-1, 3) = \left(\frac{-1 + h}{2}, \frac{0 + k}{2}\right) \]

### Step 4: Set Up the Equations
From the midpoint formula, we can set up two equations:
1. For the x-coordinates:
\[ \frac{-1 + h}{2} = -1 \]
Multiplying both sides by 2:
\[ -1 + h = -2 \implies h = -1 \]

2. For the y-coordinates:
\[ \frac{0 + k}{2} = 3 \]
Multiplying both sides by 2:
\[ k = 6 \]

### Step 5: Determine the Coordinates of Point C
Now we have the coordinates of point C:
\[ C = (-1, 6) \]

### Step 6: Find the Ordinate of Point C
The ordinate of point C is the y-coordinate, which is:
\[ \text{Ordinate of C} = 6 \]

### Final Answer
The ordinate of point C is \(6\).

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