Question

Answers are given at the end of these exercises. Write the formula for the distance \(d\) from \(P_{1}=\left(x_{1}, y_{1}\right)\) to \(P_{2}=\left(x_{2}, y_{2}\right)\)

Short answer

The distance formula is \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 } \].

Step-by-step solution

- Determine the Difference in Coordinates

Identify the coordinates for both points. Let \(P_1 = (x_1, y_1)\) and \(P_2 = (x_2, y_2)\). Calculate the difference in x-coordinates \( \text{(Δx)} = x_2 - x_1 \) and the difference in y-coordinates \( \text{(Δy)} = y_2 - y_1 \).

- Apply the Pythagorean Theorem

Using the Pythagorean theorem, the distance \(d\) between two points is given by \( d = \sqrt{(\text{Δx})^2 + (\text{Δy})^2} \). Substitute \( \text{Δx} \) and \( \text{Δy} \) from Step 1 into the formula.

- Write the Distance Formula

Expand the formula from Step 2 to the general form: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 } \]. This is the distance formula for finding the distance between two points on a plane.

Key concepts

Coordinates

Coordinates are sets of numbers that determine the position of a point on a plane. Typically, coordinates are written as \( (x, y) \), where \( x \) represents the horizontal position and \( y \) represents the vertical position. For example, the point (3, 4) means it is 3 units to the right of the origin and 4 units up.
Coordinates are fundamental in geometry and algebra because they help us locate points, compute distances, and create shapes.

In our specific exercise, we were given two points \( P_1 = (x_1, y_1) \) and \( P_2 = (x_2, y_2) \). To calculate the distance between these points, we first have to determine the difference in their x-coordinates \( \text{(Δx)} = x_2 - x_1 \) and y-coordinates \( \text{(Δy)} = y_2 - y_1 \).

Pythagorean Theorem

The Pythagorean theorem is a mathematical principle that states: in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
The theorem can be written as \( a^2 + b^2 = c^2 \). Here, \( c \) represents the hypotenuse, while \( a \) and \( b \) represent the triangle's other sides.

In the distance formula exercise, this theorem helps us find the distance between two points by treating the difference in x-coordinates and y-coordinates as legs of a right triangle.
This means if we know \( \text{Δx} \) and \( \text{Δy} \), we can apply the Pythagorean theorem: \( d = \sqrt{(\text{Δx})^2 + (\text{Δy})^2} \).
By substituting \( \text{Δx} \) and \( \text{Δy} \) from our points, we derive the distance between points.

Distance Between Points

The distance formula is used to determine the straight-line distance between two points in a coordinate plane. The formula is derived from the Pythagorean theorem and is expressed as:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 } \]
This formula provides a quick way to calculate the distance by simply plugging in the coordinates of the two points.
Here is how you can apply it:

• Identify the coordinates of the two points.
• Subtract the x-coordinate of the first point from the x-coordinate of the second point (i.e., \( \text{Δx} = x_2 - x_1 \)).
• Subtract the y-coordinate of the first point from the y-coordinate of the second point (i.e., \( \text{Δy} = y_2 - y_1 \)).
• Square both \( \text{Δx} \) and \( \text{Δy} \).
• Add the squares of \( \text{Δx} \) and \( \text{Δy} \).
• Finally, take the square root of the sum to get the distance.

This method is efficient and widely used in geometry and many real-world applications, such as navigation, to find the shortest path between two points.