3.4. Distance Between Two Points

The distance between two points P(x1,y1) and Q(x2,y2) is the length of the line segment joining them, given by the distance formula derived from the Pythagorean theorem.

This section begins by introducing the concept of the distance between two points on a coordinate plane. Points are denoted as P and Q with their respective coordinates (x1, y1) and (x2, y2). The objective is to find the length of the straight line connecting these two points. To achieve this, we use the distance formula, which is mathematically derived from the Pythagorean theorem. The formula is specifically:

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

where d represents the distance between the points, and (x1, y1) and (x2, y2) are the coordinates of the two points.

Imagine you are trying to find how far apart two friends are standing on a field. You can think of the field as a coordinate plane. If one friend is at the location (1, 2) and the other is at (4, 6), you can visualize a triangle formed by drawing lines from your friend's locations to create a right angle. Using the distance formula, you can quickly calculate how many steps it would take to get from one friend to the other.

The distance formula can be derived from the Pythagorean theorem, which applies to right triangles.

To understand the distance formula better, we need to recall the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. When finding the distance between points P and Q, we can visualize a right triangle where the lengths of the vertical and horizontal sides (the 'legs' of the triangle) are represented by the differences in x-coordinates (x2 - x1) and y-coordinates (y2 - y1). Therefore, the Pythagorean theorem gives us the relationship:

$d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$

Think of a city grid where streets run evenly in horizontal and vertical lines. If you want to walk from one corner to another, you need to walk straight up and then turn to walk over. The distance formula in the context of the city grid captures this reality. Just like calculating the diagonal shortcut across two streets, understanding the relationship between those linear paths helps you determine how far it actually is!

Using the distance formula allows you to find the length of the segment connecting any two points.

Now that we have the distance formula, we can apply it to find the distance between any two points provided in the coordinate plane. For example, if you are given points A(3, 4) and B(7, 1), you can substitute those coordinates into the distance formula:

$d = \sqrt{(7 - 3)^2 + (1 - 4)^2} = \sqrt{4 + 9} = \sqrt{13}$.

This enables you to calculate the exact distance between the points, which might be useful in various real-world applications like navigation, sports, and architecture.

Consider planning a route for a race. Knowing the starting point and the finish line on a map, you can easily calculate the distance that runners will need to cover. Using the distance formula on the coordinates of those points allows organizers to prepare better for logistics like water stations or finish line setup.

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