Find the distance between the points A `(a+b, a-b) and B (a-b,-a-b)`

Options

A. `sqrt(a^(2)+b^(2))`
B. `sqrt(2a^(2)+2b^(2))`
C. `sqrt(3a^(2)+3b^(2))`
D. `sqrt(4a^(2)+4b^(2))`

Answer

To find the distance between the points A `(a+b, a-b)` and B `(a-b, -a-b)`, we can use the distance formula. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 1: Identify the coordinates of points A and B
Let:
- Point A: \( (x_1, y_1) = (a+b, a-b) \)
- Point B: \( (x_2, y_2) = (a-b, -a-b) \)

### Step 2: Substitute the coordinates into the distance formula
Now we substitute the coordinates into the distance formula:

\[ d = \sqrt{((a-b) - (a+b))^2 + ((-a-b) - (a-b))^2} \]

### Step 3: Simplify the expression
Now we simplify each part of the formula:

1. For \(x_2 - x_1\):
\[ (a-b) - (a+b) = a - b - a - b = -2b \]

2. For \(y_2 - y_1\):
\[ (-a-b) - (a-b) = -a - b - a + b = -2a \]

### Step 4: Substitute back into the distance formula
Now we substitute these results back into the distance formula:

\[ d = \sqrt{(-2b)^2 + (-2a)^2} \]

### Step 5: Calculate the squares
Calculating the squares gives us:

\[ d = \sqrt{4b^2 + 4a^2} \]

### Step 6: Factor out the common term
We can factor out the common term:

\[ d = \sqrt{4(a^2 + b^2)} = 2\sqrt{a^2 + b^2} \]

### Final Answer
Thus, the distance between the points A and B is:

\[ d = 2\sqrt{a^2 + b^2} \]

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