Find the coordinates of the circum centre of the triangle whose vertices are (8,6) , (8,-2) and (2,-2) :

Options

A. a. (2,3)
B. b. (5,2)
C. c. (5,3)
D. d. (7,2)

Answer

To find the coordinates of the circumcenter of the triangle with vertices at \( A(8, 6) \), \( B(8, -2) \), and \( C(2, -2) \), we can follow these steps:

### Step 1: Understand the circumcenter
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. It is equidistant from all three vertices of the triangle.

### Step 2: Find the midpoints of two sides
We will find the midpoints of two sides of the triangle, say \( AB \) and \( AC \).

- **Midpoint of AB**:
\[ M_{AB} = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) = \left( \frac{8 + 8}{2}, \frac{6 + (-2)}{2} \right) = \left( 8, 2 \right) \]

- **Midpoint of AC**:
\[ M_{AC} = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{8 + 2}{2}, \frac{6 + (-2)}{2} \right) = \left( 5, 2 \right) \]

### Step 3: Find the slopes of the sides
Next, we find the slopes of the sides \( AB \) and \( AC \).

- **Slope of AB**:
\[ \text{slope}_{AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{-2 - 6}{8 - 8} = \text{undefined (vertical line)} \]

- **Slope of AC**:
\[ \text{slope}_{AC} = \frac{y_C - y_A}{x_C - x_A} = \frac{-2 - 6}{2 - 8} = \frac{-8}{-6} = \frac{4}{3} \]

### Step 4: Find the slopes of the perpendicular bisectors
The slope of the perpendicular bisector of a line is the negative reciprocal of the slope of the line.

- **Perpendicular slope of AB**:
Since \( AB \) is vertical, the perpendicular bisector will be horizontal, with a slope of \( 0 \).

- **Perpendicular slope of AC**:
\[ \text{slope}_{\text{perpendicular}} = -\frac{1}{\frac{4}{3}} = -\frac{3}{4} \]

### Step 5: Write the equations of the perpendicular bisectors
Now we can write the equations of the perpendicular bisectors.

- **Equation of the perpendicular bisector of AB** (horizontal line through \( M_{AB} \)):
\[ y = 2 \]

- **Equation of the perpendicular bisector of AC**:
Using point-slope form \( y - y_1 = m(x - x_1) \):
\[ y - 2 = -\frac{3}{4}(x - 5) \]
Simplifying this:
\[ y - 2 = -\frac{3}{4}x + \frac{15}{4} \]
\[ y = -\frac{3}{4}x + \frac{15}{4} + 2 \]
\[ y = -\frac{3}{4}x + \frac{15}{4} + \frac{8}{4} = -\frac{3}{4}x + \frac{23}{4} \]

### Step 6: Find the intersection of the two lines
To find the circumcenter, we need to solve the equations:
1. \( y = 2 \)
2. \( y = -\frac{3}{4}x + \frac{23}{4} \)

Substituting \( y = 2 \) into the second equation:
\[ 2 = -\frac{3}{4}x + \frac{23}{4} \]
Multiplying through by \( 4 \) to eliminate the fraction:
\[ 8 = -3x + 23 \]
Rearranging gives:
\[ 3x = 23 - 8 = 15 \implies x = 5 \]

### Step 7: Final coordinates of the circumcenter
Thus, the coordinates of the circumcenter are:
\[ (5, 2) \]