The triangle formed by the points A (2a, 4a), B(2a, 6a) and C `(2a +sqrt3a, 5a)` is

Options

A. right angled
B. isosceles
C. equilateral
D. None of these

Answer

To determine the type of triangle formed by the points A(2a, 4a), B(2a, 6a), and C(2a + √3a, 5a), we will calculate the lengths of the sides AB, BC, and AC using the distance formula.

### Step-by-Step Solution:

1. **Identify the Points:**
- A = (2a, 4a)
- B = (2a, 6a)
- C = (2a + √3a, 5a)

2. **Calculate the Length of Side AB:**
- The distance formula between two points (x1, y1) and (x2, y2) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- For points A and B:
\[ AB = \sqrt{(2a - 2a)^2 + (6a - 4a)^2} = \sqrt{0 + (2a)^2} = \sqrt{4a^2} = 2a \]

3. **Calculate the Length of Side BC:**
- For points B and C:
\[ BC = \sqrt{(2a + \sqrt{3}a - 2a)^2 + (5a - 6a)^2} \]
\[ = \sqrt{(\sqrt{3}a)^2 + (-a)^2} = \sqrt{3a^2 + a^2} = \sqrt{4a^2} = 2a \]

4. **Calculate the Length of Side AC:**
- For points A and C:
\[ AC = \sqrt{(2a + \sqrt{3}a - 2a)^2 + (5a - 4a)^2} \]
\[ = \sqrt{(\sqrt{3}a)^2 + (1a)^2} = \sqrt{3a^2 + a^2} = \sqrt{4a^2} = 2a \]

5. **Conclusion:**
- We have found that:
- AB = 2a
- BC = 2a
- AC = 2a
- Since all three sides are equal, the triangle formed by points A, B, and C is an **equilateral triangle**.

### Final Answer:
The triangle formed by the points A(2a, 4a), B(2a, 6a), and C(2a + √3a, 5a) is an **equilateral triangle**.