For triangele PQR , find equation of altitude PS if co-ordinates of P, Q and R are (1, 2) , (2, -1) and (0, 5) respectively ?

Options

A. x - 3y = - 7
B. x + 3 y = - 5
C. x - 3y= - 5
D. x + 3y = - 7

Answer

To find the equation of the altitude PS of triangle PQR, given the coordinates of the points P(1, 2), Q(2, -1), and R(0, 5), we will follow these steps:

### Step 1: Find the slope of line QR
The slope (m) of a line through two points (x1, y1) and (x2, y2) is given by the formula:
\[ m = \frac{y2 - y1}{x2 - x1} \]

For points Q(2, -1) and R(0, 5):
- \( x1 = 2, y1 = -1 \)
- \( x2 = 0, y2 = 5 \)

Calculating the slope:
\[ m_{QR} = \frac{5 - (-1)}{0 - 2} = \frac{5 + 1}{0 - 2} = \frac{6}{-2} = -3 \]

### Step 2: Find the slope of the altitude PS
The slope of the altitude PS will be the negative reciprocal of the slope of QR. If the slope of QR is -3, then the slope of PS (m_PS) is:
\[ m_{PS} = -\frac{1}{m_{QR}} = -\frac{1}{-3} = \frac{1}{3} \]

### Step 3: Use the point-slope form to find the equation of line PS
The point-slope form of a line is given by:
\[ y - y1 = m(x - x1) \]
Where (x1, y1) is a point on the line and m is the slope.

Using point P(1, 2) and the slope \( \frac{1}{3} \):
\[ y - 2 = \frac{1}{3}(x - 1) \]

### Step 4: Simplify the equation
Now, we will simplify the equation:
1. Distributing the slope:
\[ y - 2 = \frac{1}{3}x - \frac{1}{3} \]

2. Adding 2 to both sides:
\[ y = \frac{1}{3}x - \frac{1}{3} + 2 \]
\[ y = \frac{1}{3}x - \frac{1}{3} + \frac{6}{3} \]
\[ y = \frac{1}{3}x + \frac{5}{3} \]

### Step 5: Convert to standard form
To convert this to standard form (Ax + By = C), we can multiply through by 3 to eliminate the fraction:
\[ 3y = x + 5 \]
Rearranging gives:
\[ x - 3y + 5 = 0 \]
Or:
\[ x - 3y = -5 \]

### Final Answer
The equation of the altitude PS is:
\[ x - 3y = -5 \]