If point \( A \) is located at coordinates \( (5,3) \) and point \( B \) is located at coordinates \( (-3,9) \), what is the distance from \( A \) to \( B \) if the units of the coordinated system are meters?

To find the distance between points \( A(5,3) \) and \( B(-3,9) \), you can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates of points \( A \) and \( B \): \[ d = \sqrt{((-3) - 5)^2 + (9 - 3)^2} \] Calculating this step by step: 1. \( -3 - 5 = -8 \) 2. \( 9 - 3 = 6 \) 3. Now, substitute into the formula: \[ d = \sqrt{(-8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \] So, the distance from point \( A \) to point \( B \) is \( 10 \) meters. When you delve into the history of the distance formula, it connects back to Pythagoras, who discovered that the square of the hypotenuse in a right triangle is equal to the sum of the squares of the other two sides. This ancient mathematical concept has been foundational in both geometry and algebra, providing a framework still in use today! In real-world scenarios, knowing how to calculate distances can be incredibly handy. Think about urban planning, navigation, or even setting up a sound system for an event – knowing how far apart elements need to be can save a lot of hassle. Whether you’re measuring the distance between buildings or figuring out the best layout for a concert, this formula helps you get it right!