Points A(6,6) ,B(2,3) and C(4,7) are the vertices of a triangle which is :

Options

A. right angled
B. acute angled
C. obtuse angled
D. none of these

Answer

To determine the type of triangle formed by the points A(6, 6), B(2, 3), and C(4, 7), we can follow these steps:

### Step 1: Calculate the lengths of the sides of the triangle.
We will use the distance formula to find the lengths of the sides AB, BC, and CA.

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

#### 1.1 Calculate AB:
\[ AB = \sqrt{(2 - 6)^2 + (3 - 6)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

#### 1.2 Calculate BC:
\[ BC = \sqrt{(4 - 2)^2 + (7 - 3)^2} = \sqrt{(2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]

#### 1.3 Calculate CA:
\[ CA = \sqrt{(6 - 4)^2 + (6 - 7)^2} = \sqrt{(2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \]

### Step 2: Determine the type of triangle using the lengths of the sides.
To classify the triangle, we can use the property of the angles based on the lengths of the sides. We will apply the triangle inequality and the Pythagorean theorem.

#### 2.1 Check for a right triangle:
For a triangle with sides \(a\), \(b\), and \(c\) (where \(c\) is the longest side), it is a right triangle if:
\[ c^2 = a^2 + b^2 \]

Here, we have:
- \(AB = 5\)
- \(BC = 2\sqrt{5}\)
- \(CA = \sqrt{5}\)

Let's identify the longest side:
- \(AB = 5\) (longest side)
- \(BC = 2\sqrt{5} \approx 4.47\)
- \(CA = \sqrt{5} \approx 2.24\)

Now, we check:
\[ AB^2 = 5^2 = 25 \]
\[ BC^2 + CA^2 = (2\sqrt{5})^2 + (\sqrt{5})^2 = 20 + 5 = 25 \]

Since \(AB^2 = BC^2 + CA^2\), the triangle is a right triangle.

### Conclusion:
The triangle formed by the points A(6, 6), B(2, 3), and C(4, 7) is a right triangle.