The orthocentre of ` triangle ABC` where vertices are A(-1,7), B(-7,1), C(5,-5) is

Options

A. -3,3
B. 3,-3
C. 3,3
D. -3,-3

Answer

To find the orthocenter of triangle ABC with vertices A(-1, 7), B(-7, 1), and C(5, -5), we will follow these steps:

### Step 1: Find the slopes of the sides of the triangle.
1. **Slope of BC**:
- Points B(-7, 1) and C(5, -5).
- Slope (m) = (y2 - y1) / (x2 - x1) = (-5 - 1) / (5 - (-7)) = (-6) / (12) = -1/2.

2. **Slope of AC**:
- Points A(-1, 7) and C(5, -5).
- Slope = (-5 - 7) / (5 - (-1)) = (-12) / (6) = -2.

3. **Slope of AB**:
- Points A(-1, 7) and B(-7, 1).
- Slope = (1 - 7) / (-7 - (-1)) = (-6) / (-6) = 1.

### Step 2: Find the slopes of the altitudes.
1. **Slope of altitude from A to BC (AD)**:
- Since AD is perpendicular to BC, the slope of AD = -1 / (slope of BC) = -1 / (-1/2) = 2.

2. **Slope of altitude from B to AC (BE)**:
- Slope of BE = -1 / (slope of AC) = -1 / (-2) = 1/2.

### Step 3: Write the equations of the altitudes.
1. **Equation of altitude AD** (passing through A(-1, 7)):
- Using point-slope form: y - y1 = m(x - x1).
- y - 7 = 2(x + 1).
- y - 7 = 2x + 2.
- y = 2x + 9. (Equation 1)

2. **Equation of altitude BE** (passing through B(-7, 1)):
- y - 1 = (1/2)(x + 7).
- y - 1 = (1/2)x + (7/2).
- y = (1/2)x + (9/2). (Equation 2)

### Step 4: Solve the equations of the altitudes to find the orthocenter.
1. Set Equation 1 equal to Equation 2:
- 2x + 9 = (1/2)x + (9/2).
- Multiply through by 2 to eliminate the fraction:
- 4x + 18 = x + 9.
- 4x - x = 9 - 18.
- 3x = -9.
- x = -3.

2. Substitute x = -3 into Equation 1 to find y:
- y = 2(-3) + 9 = -6 + 9 = 3.

### Conclusion:
The orthocenter of triangle ABC is at the point (-3, 3).

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