Two vertices of a triangle ABC are B(5,-1) and C(-2,3) .If the orthocentre of the triangle is the origin , find the thrid vertex.

Options

A. a. `(7/2 , (13)/2)`
B. `(3/2 , (11)/(2))`
C. `(-4,-7)`
D. none of these

Answer

To find the third vertex \( A(x, y) \) of triangle \( ABC \) given the vertices \( B(5, -1) \) and \( C(-2, 3) \) with the orthocenter at the origin \( O(0, 0) \), we can follow these steps:

### Step 1: Understand the properties of the orthocenter
The orthocenter of a triangle is the point where the altitudes intersect. Since the orthocenter is at the origin, the altitudes from points \( A \), \( B \), and \( C \) will pass through the origin.

### Step 2: Find the slope of the altitude from point \( B \)
The slope of the line \( BC \) can be calculated using the formula:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
For points \( B(5, -1) \) and \( C(-2, 3) \):
\[ \text{slope of } BC = \frac{3 - (-1)}{-2 - 5} = \frac{4}{-7} = -\frac{4}{7} \]
The slope of the altitude from \( B \) to line \( AC \) will be the negative reciprocal of the slope of \( BC \):
\[ \text{slope of altitude from } B = \frac{7}{4} \]

### Step 3: Write the equation of the altitude from point \( B \)
Using point-slope form \( y - y_1 = m(x - x_1) \):
\[ y - (-1) = \frac{7}{4}(x - 5) \]
This simplifies to:
\[ y + 1 = \frac{7}{4}x - \frac{35}{4} \]
\[ y = \frac{7}{4}x - \frac{39}{4} \quad \text{(Equation 1)} \]

### Step 4: Find the slope of the altitude from point \( C \)
Using the same method, we find the slope of the line \( AB \) using points \( A(x, y) \) and \( B(5, -1) \):
The slope of the altitude from \( C \) to line \( AB \) will be the negative reciprocal of the slope of \( AB \).

### Step 5: Write the equation of the altitude from point \( C \)
Using the slope from point \( C(-2, 3) \):
\[ y - 3 = \text{slope of } AB (x + 2) \]
This will also give us another equation.

### Step 6: Solve the equations simultaneously
Now we have two equations (from the altitudes):
1. \( y = \frac{7}{4}x - \frac{39}{4} \)
2. The equation from the altitude from \( C \).

By substituting the value of \( y \) from Equation 1 into the second equation, we can solve for \( x \) and then find \( y \).

### Step 7: Find the coordinates of point \( A \)
After solving the equations, we will find the coordinates of point \( A \).

### Final Result
The coordinates of the third vertex \( A \) will be \( (-4, -7) \).

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