What is the perimeter of triangle with vertices (0,0)(1,0)and(0,1).

Answer

To find the perimeter of the triangle with vertices at (0,0), (1,0), and (0,1), we can follow these steps:

### Step 1: Identify the vertices
The vertices of the triangle are:
- \( O(0, 0) \)
- \( A(1, 0) \)
- \( B(0, 1) \)

### Step 2: Calculate the lengths of the sides
We need to find the lengths of the sides \( OA \), \( OB \), and \( AB \).

1. **Length of \( OA \)**:
- The distance between points \( O(0, 0) \) and \( A(1, 0) \) is calculated as:
\[ OA = \sqrt{(1 - 0)^2 + (0 - 0)^2} = \sqrt{1^2 + 0^2} = \sqrt{1} = 1 \]

2. **Length of \( OB \)**:
- The distance between points \( O(0, 0) \) and \( B(0, 1) \) is calculated as:
\[ OB = \sqrt{(0 - 0)^2 + (1 - 0)^2} = \sqrt{0^2 + 1^2} = \sqrt{1} = 1 \]

3. **Length of \( AB \)**:
- The distance between points \( A(1, 0) \) and \( B(0, 1) \) can be calculated using the Pythagorean theorem:
\[ AB = \sqrt{(0 - 1)^2 + (1 - 0)^2} = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Calculate the perimeter
The perimeter \( P \) of the triangle is the sum of the lengths of its sides:
\[ P = OA + OB + AB = 1 + 1 + \sqrt{2} \]
Thus, the perimeter is:
\[ P = 2 + \sqrt{2} \]

### Final Answer
The perimeter of the triangle is \( 2 + \sqrt{2} \) units.

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