A (1, -5), B (2, 2) and C (-2, 4) are the vertices of triangle ABC. find the equation of:
the altitude of the triangle through B.

Answer

To find the equation of the altitude of triangle ABC through vertex B, we will follow these steps:

### Step 1: Identify the coordinates of the vertices
The vertices of the triangle are given as:
- A(1, -5)
- B(2, 2)
- C(-2, 4)

### Step 2: Find the slope of line AC
To find the slope of line AC, we use the formula for the slope between two points (y2 - y1) / (x2 - x1).

Let:
- A(1, -5) = (x1, y1)
- C(-2, 4) = (x2, y2)

Calculating the slope (m_AC):
\[ m_{AC} = \frac{y2 - y1}{x2 - x1} = \frac{4 - (-5)}{-2 - 1} = \frac{4 + 5}{-3} = \frac{9}{-3} = -3 \]

### Step 3: Determine the slope of the altitude BD
Since the altitude BD is perpendicular to line AC, the product of their slopes (m_BD and m_AC) is -1.
\[ m_{BD} \cdot m_{AC} = -1 \implies m_{BD} \cdot (-3) = -1 \]
Solving for m_BD:
\[ m_{BD} = \frac{1}{3} \]

### Step 4: Use point-slope form to find the equation of line BD
Using the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
where (x1, y1) is point B(2, 2) and m is the slope we found (1/3):
\[ y - 2 = \frac{1}{3}(x - 2) \]

### Step 5: Simplify the equation
Distributing the slope on the right side:
\[ y - 2 = \frac{1}{3}x - \frac{2}{3} \]
Now, add 2 to both sides:
\[ y = \frac{1}{3}x - \frac{2}{3} + 2 \]
Convert 2 to a fraction with a denominator of 3:
\[ y = \frac{1}{3}x - \frac{2}{3} + \frac{6}{3} \]
Combine the fractions:
\[ y = \frac{1}{3}x + \frac{4}{3} \]

### Step 6: Rearrange to standard form
To express it in standard form (Ax + By + C = 0), we can rearrange:
\[ -\frac{1}{3}x + y - \frac{4}{3} = 0 \]
Multiplying through by 3 to eliminate the fraction:
\[ -x + 3y - 4 = 0 \]
or
\[ x - 3y + 4 = 0 \]

### Final Answer
The equation of the altitude BD through vertex B is:
\[ x - 3y + 4 = 0 \]

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