Consider a triangle PQR with coordinates of its vertices as P(-8,5), Q(-15, -19), and R (1, -7). The bisector of the interior angle of P has the equation which can be written in the form ax+2y+c=0.
The distance between the orthocenter and the circumcenter of triangle PQR is

Options

A. `25//2`
B. `29//2`
C. `37//2`
D. `51//2`

Explanation

To solve the problem step by step, we will first find the equation of the angle bisector of angle P and then calculate the distance between the orthocenter and circumcenter of triangle PQR.

### Step 1: Identify the coordinates of the vertices
The coordinates of the vertices of triangle PQR are:
- P(-8, 5)
- Q(-15, -19)
- R(1, -7)

### Step 2: Calculate the slopes of sides PQ and PR
To find the equation of the angle bisector, we first need the slopes of the sides PQ and PR.

1. **Slope of PQ**:
\[ m_{PQ} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-19 - 5}{-15 - (-8)} = \frac{-24}{-7} = \frac{24}{7} \]

2. **Slope of PR**:
\[ m_{PR} = \frac{-7 - 5}{1 - (-8)} = \frac{-12}{9} = -\frac{4}{3} \]

### Step 3: Find the angle bisector using the formula
The angle bisector can be found using the formula for the angle bisector in terms of slopes:
\[ \frac{y - y_1}{x - x_1} = \frac{m_1 + m_2}{1 - m_1 m_2} \]
where \(m_1\) and \(m_2\) are the slopes of the lines.

Using the coordinates of point P(-8, 5):
\[ \frac{y - 5}{x + 8} = \frac{\frac{24}{7} - \frac{4}{3}}{1 + \frac{24}{7} \cdot \left(-\frac{4}{3}\right)} \]

Calculating the numerator:
\[ \frac{24}{7} - \frac{4}{3} = \frac{72 - 28}{21} = \frac{44}{21} \]

Calculating the denominator:
\[ 1 - \left(\frac{24}{7} \cdot -\frac{4}{3}\right) = 1 + \frac{96}{21} = \frac{21 + 96}{21} = \frac{117}{21} \]

Thus, the equation of the angle bisector becomes:
\[ \frac{y - 5}{x + 8} = \frac{44/21}{117/21} = \frac{44}{117} \]

Cross-multiplying gives:
\[ 117(y - 5) = 44(x + 8) \]
Expanding and rearranging:
\[ 117y - 585 = 44x + 352 \implies 44x - 117y + 937 = 0 \]

### Step 4: Convert to the required form
We need to express this in the form \(ax + 2y + c = 0\). We can divide the entire equation by 2:
\[ 22x - \frac{117}{2}y + \frac{937}{2} = 0 \]
This is not in the required form, so we will keep it as is for now.

### Step 5: Find the circumcenter
In a right triangle, the circumcenter is the midpoint of the hypotenuse. The hypotenuse is QR.

1. **Midpoint M of QR**:
\[ M = \left(\frac{-15 + 1}{2}, \frac{-19 + (-7)}{2}\right) = \left(\frac{-14}{2}, \frac{-26}{2}\right) = (-7, -13) \]

### Step 6: Identify the orthocenter
In a right triangle, the orthocenter is the vertex at the right angle. Here, R(1, -7) is the orthocenter.

### Step 7: Calculate the distance between orthocenter R and circumcenter M
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of R(1, -7) and M(-7, -13):
\[ d = \sqrt{(-7 - 1)^2 + (-13 + 7)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

### Final Answer
The distance between the orthocenter and the circumcenter of triangle PQR is **10 units**.

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