The equation of the sides of a triangle, the coordinates of whose angular points are: `(3,5),(1,2)` and `(-7,4)`.

Options

A. `3x-2y+1=0`
B. `x-10y+47=0`
C. `x+4y-9=0`
D. All of these

Answer

To find the equations of the sides of the triangle with vertices at the points A(3, 5), B(1, 2), and C(-7, 4), we will use the two-point form of the equation of a line. The two-point form is given by:

\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \]

### Step 1: Find the equation of line AB

1. **Identify the coordinates of points A and B**:
- A(3, 5) → \(x_1 = 3, y_1 = 5\)
- B(1, 2) → \(x_2 = 1, y_2 = 2\)

2. **Substitute into the two-point form**:
\[ y - 5 = \frac{2 - 5}{1 - 3} (x - 3) \]

3. **Calculate the slope**:
\[ \frac{2 - 5}{1 - 3} = \frac{-3}{-2} = \frac{3}{2} \]

4. **Substitute the slope back into the equation**:
\[ y - 5 = \frac{3}{2} (x - 3) \]

5. **Multiply both sides by 2 to eliminate the fraction**:
\[ 2(y - 5) = 3(x - 3) \]

6. **Distribute**:
\[ 2y - 10 = 3x - 9 \]

7. **Rearrange to standard form**:
\[ 3x - 2y + 1 = 0 \]

### Step 2: Find the equation of line BC

1. **Identify the coordinates of points B and C**:
- B(1, 2) → \(x_1 = 1, y_1 = 2\)
- C(-7, 4) → \(x_2 = -7, y_2 = 4\)

2. **Substitute into the two-point form**:
\[ y - 2 = \frac{4 - 2}{-7 - 1} (x - 1) \]

3. **Calculate the slope**:
\[ \frac{4 - 2}{-7 - 1} = \frac{2}{-8} = -\frac{1}{4} \]

4. **Substitute the slope back into the equation**:
\[ y - 2 = -\frac{1}{4} (x - 1) \]

5. **Multiply both sides by 4**:
\[ 4(y - 2) = -(x - 1) \]

6. **Distribute**:
\[ 4y - 8 = -x + 1 \]

7. **Rearrange to standard form**:
\[ x + 4y - 9 = 0 \]

### Step 3: Find the equation of line AC

1. **Identify the coordinates of points A and C**:
- A(3, 5) → \(x_1 = 3, y_1 = 5\)
- C(-7, 4) → \(x_2 = -7, y_2 = 4\)

2. **Substitute into the two-point form**:
\[ y - 5 = \frac{4 - 5}{-7 - 3} (x - 3) \]

3. **Calculate the slope**:
\[ \frac{4 - 5}{-7 - 3} = \frac{-1}{-10} = \frac{1}{10} \]

4. **Substitute the slope back into the equation**:
\[ y - 5 = \frac{1}{10} (x - 3) \]

5. **Multiply both sides by 10**:
\[ 10(y - 5) = x - 3 \]

6. **Distribute**:
\[ 10y - 50 = x - 3 \]

7. **Rearrange to standard form**:
\[ x - 10y + 47 = 0 \]

### Summary of the Equations of the Sides of the Triangle:
- Equation of line AB: \(3x - 2y + 1 = 0\)
- Equation of line BC: \(x + 4y - 9 = 0\)
- Equation of line AC: \(x - 10y + 47 = 0\)