If ABC is a triangle such that A=(1,2) and B=(5,5) with BC=9 and AC =12 units , then slope of altitude CD is (D is a point on BC)

Options

A. `-4/3`
B. `4/3`
C. `3/4`
D. `-3/4`

Answer

To find the slope of the altitude CD in triangle ABC, we will follow these steps:

### Step 1: Identify the coordinates of points A and B
Given:
- A = (1, 2)
- B = (5, 5)

### Step 2: Calculate the slope of line AB
The formula for the slope (m) between two points (x1, y1) and (x2, y2) is:
\[ m = \frac{y2 - y1}{x2 - x1} \]

For points A and B:
- \( y2 = 5, y1 = 2, x2 = 5, x1 = 1 \)

Substituting these values:
\[ m_{AB} = \frac{5 - 2}{5 - 1} = \frac{3}{4} \]

### Step 3: Determine the slope of altitude CD
Since CD is the altitude from point C to line AB, it is perpendicular to AB. The product of the slopes of two perpendicular lines is -1. Therefore:
\[ m_{AB} \cdot m_{CD} = -1 \]

Substituting the slope of AB:
\[ \frac{3}{4} \cdot m_{CD} = -1 \]

### Step 4: Solve for the slope of CD
To find \( m_{CD} \):
\[ m_{CD} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{3}{4}} = -\frac{4}{3} \]

### Conclusion
The slope of altitude CD is:
\[ m_{CD} = -\frac{4}{3} \]