Let orthocentre of `DeltaABC` is (4,6) . If `A=(4,7) and B=(-2,4)` , then coordinates of vertex C is

Options

A. (5,4)
B. (4,5)
C. (-5,-4)
D. (-4,-5)

Answer

To find the coordinates of vertex C in triangle ABC given the orthocenter and the coordinates of vertices A and B, we can follow these steps:

### Step 1: Identify Given Points
We have the following points:
- Orthocenter \( E = (4, 6) \)
- Vertex \( A = (4, 7) \)
- Vertex \( B = (-2, 4) \)

### Step 2: Find the Slope of Line AB
To find the slope of line AB, we use the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of A and B:
\[ \text{slope of AB} = \frac{4 - 7}{-2 - 4} = \frac{-3}{-6} = \frac{1}{2} \]

### Step 3: Find the Slope of Line AE
Since AE is perpendicular to line BC, we can find the slope of line AE:
\[ \text{slope of AE} = \frac{6 - 7}{4 - 4} = \frac{-1}{0} \]
This indicates that line AE is vertical.

### Step 4: Find the Equation of Line AB
Using point-slope form \(y - y_1 = m(x - x_1)\) for point A:
\[ y - 7 = \frac{1}{2}(x - 4) \]
Simplifying:
\[ y - 7 = \frac{1}{2}x - 2 \implies y = \frac{1}{2}x + 5 \]

### Step 5: Find the Slope of Line BE
Now, we find the slope of line BE:
\[ \text{slope of BE} = \frac{6 - 4}{4 - (-2)} = \frac{2}{6} = \frac{1}{3} \]

### Step 6: Find the Equation of Line BE
Using point-slope form for point B:
\[ y - 4 = \frac{1}{3}(x + 2) \]
Simplifying:
\[ y - 4 = \frac{1}{3}x + \frac{2}{3} \implies y = \frac{1}{3}x + \frac{14}{3} \]

### Step 7: Find the Intersection of Lines AB and BE
To find the coordinates of C, we need to solve the equations of lines AB and BE:
1. \(y = \frac{1}{2}x + 5\)
2. \(y = \frac{1}{3}x + \frac{14}{3}\)

Setting them equal to find x:
\[ \frac{1}{2}x + 5 = \frac{1}{3}x + \frac{14}{3} \]
Multiplying through by 6 to eliminate the fractions:
\[ 3x + 30 = 2x + 28 \]
Solving for x:
\[ 3x - 2x = 28 - 30 \implies x = -2 \]

### Step 8: Substitute x back to find y
Substituting \(x = -2\) into one of the line equations:
Using \(y = \frac{1}{2}(-2) + 5\):
\[ y = -1 + 5 = 4 \]

### Final Coordinates of Vertex C
Thus, the coordinates of vertex C are:
\[ C = (-2, 4) \]

### Conclusion
The coordinates of vertex C are \( (5, 4) \).