Find the distance between the following pairs of points:
(1) \(2\sqrt{2}\)
(2) \(4\sqrt{2}\)
(3) \(2\sqrt{a^2 + b^2}\)
The given points are \((2, 3)\) and \((4, 1)\).
We use the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
Substituting the values: \( d = \sqrt{(4 - 2)^2 + (1 - 3)^2} \).
This becomes: \( d = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} \).
Thus, the distance is \( d = 2\sqrt{2} \).
The given points are \((-5, 7)\) and \((-1, 3)\).
Apply the distance formula: \( d = \sqrt{(-1 + 5)^2 + (3 - 7)^2} \).
Simplifying inside the brackets gives: \( d = \sqrt{4^2 + (-4)^2} \).
Then: \( d = \sqrt{16 + 16} = \sqrt{32} \).
So, the distance is \( d = 4\sqrt{2} \).
The given points are \((a, b)\) and \((-a, -b)\).
Using the formula: \( d = \sqrt{(-a - a)^2 + (-b - b)^2} \).
This simplifies to: \( d = \sqrt{(-2a)^2 + (-2b)^2} \).
Compute squares: \( (-2a)^2 = 4a^2 \) and \( (-2b)^2 = 4b^2 \).
Thus: \( d = \sqrt{4a^2 + 4b^2} = \sqrt{4(a^2 + b^2)} \).
Therefore, the distance is \( d = 2\sqrt{a^2 + b^2} \).