Consider triangle ABC where `A=(4,4),B=(7,4),C=(4,7)` then

Options

A. The centriod of triangle ABC is (5,5)
B. The orthocentre of triangle ABC is (4,4)
C. The circum centre is `((11)/(2),(11)/(2))`
D. The incentre is `((11+4sqrt2)/(2+sqrt2),(11+4sqrt2)/(2+sqrt2))`

Answer

To solve the problem, we need to find the centroid, orthocenter, circumcenter, and incenter of triangle ABC with the given vertices A(4, 4), B(7, 4), and C(4, 7).

### Step 1: Finding the Centroid
The formula for the centroid \( G \) of a triangle with vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is given by:

\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

Substituting the coordinates of points A, B, and C:

\[ G = \left( \frac{4 + 7 + 4}{3}, \frac{4 + 4 + 7}{3} \right) = \left( \frac{15}{3}, \frac{15}{3} \right) = (5, 5) \]

### Step 2: Finding the Orthocenter
For a right triangle, the orthocenter is located at the vertex where the right angle is formed. In triangle ABC, the right angle is at vertex A(4, 4).

Thus, the orthocenter \( H \) is:

\[ H = (4, 4) \]

### Step 3: Finding the Circumcenter
The circumcenter of a right triangle is the midpoint of the hypotenuse. The hypotenuse is BC, with points B(7, 4) and C(4, 7).

The midpoint \( O \) is calculated as:

\[ O = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2} \right) = \left( \frac{7 + 4}{2}, \frac{4 + 7}{2} \right) = \left( \frac{11}{2}, \frac{11}{2} \right) \]

### Step 4: Finding the Incenter
The incenter \( I \) of a triangle can be found using the formula:

\[ I = \left( \frac{a x_1 + b x_2 + c x_3}{a + b + c}, \frac{a y_1 + b y_2 + c y_3}{a + b + c} \right) \]

Where \( a, b, c \) are the lengths of the sides opposite to vertices \( A, B, C \) respectively.

1. Calculate the lengths of the sides:
- \( a = BC = \sqrt{(7 - 4)^2 + (4 - 7)^2} = \sqrt{3^2 + (-3)^2} = \sqrt{18} = 3\sqrt{2} \)
- \( b = AC = \sqrt{(4 - 4)^2 + (7 - 4)^2} = \sqrt{0 + 3^2} = 3 \)
- \( c = AB = \sqrt{(7 - 4)^2 + (4 - 4)^2} = \sqrt{3^2 + 0} = 3 \)

2. Substitute into the incenter formula:
\[ I = \left( \frac{(3\sqrt{2}) \cdot 4 + 3 \cdot 7 + 3 \cdot 4}{3\sqrt{2} + 3 + 3}, \frac{(3\sqrt{2}) \cdot 4 + 3 \cdot 4 + 3 \cdot 7}{3\sqrt{2} + 3 + 3} \right) \]

Simplifying:
\[ I = \left( \frac{12\sqrt{2} + 21}{3\sqrt{2} + 6}, \frac{12\sqrt{2} + 21}{3\sqrt{2} + 6} \right) \]

This results in:
\[ I = \left( \frac{11 + 4\sqrt{2}}{2 + \sqrt{2}}, \frac{11 + 4\sqrt{2}}{2 + \sqrt{2}} \right) \]

### Summary of Results
- Centroid \( G = (5, 5) \)
- Orthocenter \( H = (4, 4) \)
- Circumcenter \( O = \left( \frac{11}{2}, \frac{11}{2} \right) \)
- Incenter \( I = \left( \frac{11 + 4\sqrt{2}}{2 + \sqrt{2}}, \frac{11 + 4\sqrt{2}}{2 + \sqrt{2}} \right) \)