The points (2, 2) (6, 3) and (4, 11) are the vertices of

Options

A. an equilateral
B. a right angled triangle
C. an isosceles triangle
D. a scalene triangle

Answer

To determine the type of triangle formed by the points (2, 2), (6, 3), and (4, 11), we will follow these steps:

### Step 1: Assign the Points
Let:
- Point A = (2, 2)
- Point B = (6, 3)
- Point C = (4, 11)

### Step 2: Calculate the Distances Between the Points
We will use the distance formula:
\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

#### Distance AB
Using points A and B:
\[ AB = \sqrt{(6 - 2)^2 + (3 - 2)^2} \]
\[ = \sqrt{(4)^2 + (1)^2} \]
\[ = \sqrt{16 + 1} \]
\[ = \sqrt{17} \]

#### Distance BC
Using points B and C:
\[ BC = \sqrt{(4 - 6)^2 + (11 - 3)^2} \]
\[ = \sqrt{(-2)^2 + (8)^2} \]
\[ = \sqrt{4 + 64} \]
\[ = \sqrt{68} \]

#### Distance AC
Using points A and C:
\[ AC = \sqrt{(4 - 2)^2 + (11 - 2)^2} \]
\[ = \sqrt{(2)^2 + (9)^2} \]
\[ = \sqrt{4 + 81} \]
\[ = \sqrt{85} \]

### Step 3: Analyze the Triangle
Now we have the lengths of the sides:
- AB = \( \sqrt{17} \)
- BC = \( \sqrt{68} \)
- AC = \( \sqrt{85} \)

### Step 4: Check for Right-Angled Triangle
To check if the triangle is a right-angled triangle, we can use the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
where \( c \) is the longest side.

Here, \( AC \) is the longest side:
- \( AC^2 = 85 \)
- \( AB^2 = 17 \)
- \( BC^2 = 68 \)

Now, check if:
\[ AC^2 = AB^2 + BC^2 \]
\[ 85 = 17 + 68 \]
\[ 85 = 85 \]

Since this is true, the triangle is a right-angled triangle.

### Conclusion
The points (2, 2), (6, 3), and (4, 11) are the vertices of a right-angled triangle.

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