A, B and C have co-ordinates (0, 3), (4, 4) and (8, 0) of triangle ABC respectively. Find the equation of the line through A and perpendicular to BC. 

Answer

To find the equation of the line through point A (0, 3) and perpendicular to line BC, we will follow these steps:

### Step 1: Identify the coordinates of points B and C
The coordinates of points B and C are given as:
- B = (4, 4)
- C = (8, 0)

### Step 2: Calculate the slope of line BC
The slope (m) of a line through two points (x1, y1) and (x2, y2) is calculated using the formula:
\[ m = \frac{y2 - y1}{x2 - x1} \]

For points B (4, 4) and C (8, 0):
- \( x1 = 4, y1 = 4 \)
- \( x2 = 8, y2 = 0 \)

Substituting these values into the slope formula:
\[ m_{BC} = \frac{0 - 4}{8 - 4} = \frac{-4}{4} = -1 \]

### Step 3: Determine the slope of the line through A that is perpendicular to BC
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. Therefore, if the slope of BC (m_{BC}) is -1, the slope of the line through A (m_{AM}) will be:
\[ m_{AM} = -\frac{1}{m_{BC}} = -\frac{1}{-1} = 1 \]

### Step 4: Use the point-slope form to find the equation of the line through A
The point-slope form of the equation of a line is given by:
\[ y - y1 = m(x - x1) \]

Here, we have:
- Point A (x1, y1) = (0, 3)
- Slope (m) = 1

Substituting these values into the point-slope form:
\[ y - 3 = 1(x - 0) \]

### Step 5: Simplify the equation
Now, we simplify the equation:
\[ y - 3 = x \]
\[ y = x + 3 \]

### Final Answer
The equation of the line through point A and perpendicular to line BC is:
\[ y = x + 3 \]

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