Find the distance between the two points (a+b,a-b) and (b-a,a+b).

To find the distance between the two points \((a + b, a - b)\) and \((b - a, a + b)\), we use the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\):

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

Here, the coordinates of the two points are:
- \((x_1, y_1) = (a + b, a - b)\)
- \((x_2, y_2) = (b - a, a + b)\)

Now, apply the distance formula:

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

### Step 1: Simplify each component
1. \( (b - a) - (a + b) = b - a - a - b = -2a \)
2. \( (a + b) - (a - b) = a + b - a + b = 2b \)

### Step 2: Substitute back into the distance formula
\[
\text{Distance} = \sqrt{(-2a)^2 + (2b)^2}
\]
\[
\text{Distance} = \sqrt{4a^2 + 4b^2}
\]
\[
\text{Distance} = \sqrt{4(a^2 + b^2)}
\]
\[
\text{Distance} = 2\sqrt{a^2 + b^2}
\]

Thus, the distance between the two points \((a + b, a - b)\) and \((b - a, a + b)\) is:

\[
\boxed{2\sqrt{a^2 + b^2}}
\]

Would you like more details on any step or have any questions?

Here are five related questions to further explore this topic:

1. How would the distance formula change in three-dimensional space?
2. Can you find the midpoint of the line segment connecting \((a + b, a - b)\) and \((b - a, a + b)\)?
3. What would be the distance between two points on a circle of radius \(r\)?
4. How can the distance formula be used to derive the equation of a circle?
5. How does the distance formula relate to the Pythagorean Theorem?

**Tip:** Memorizing the distance formula and recognizing its similarity to the Pythagorean Theorem can simplify problems involving distance in geometry.

Find the distance between the two points (a+b, a-b) and (b-a, a+b).

Learn how to find the distance between two points with coordinates (a+b, a-b) and (b-a, a+b) using the distance formula. This detailed step-by-step solution includes simplifying expressions and applying the Pythagorean Theorem, suitable for students in Grades 9-12.

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