Question

Find the distance \(d\) between the points \(P_{1}\) and \(P_{2}\). $$ P_{1}=(-0.2,0.3) ; \quad P_{2}=(2.3,1.1) $$

Short answer

The distance is approximately 2.626 units.

Step-by-step solution

Understand the Distance Formula

The distance formula for two points \( P_{1}=(x_{1}, y_{1}) \) and \( P_{2}=(x_{2}, y_{2}) \) in a coordinate plane is given by \[ d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}} \]. This formula derives from the Pythagorean theorem.

Identify the Coordinates

Assign the coordinates from the problem to the variables in the distance formula: \( P_{1}=(-0.2,0.3) \) so \( x_{1} = -0.2 \) and \( y_{1} = 0.3 \). Also, \( P_{2}=(2.3,1.1) \) so \( x_{2} = 2.3 \) and \( y_{2} = 1.1 \).

Substitute the Coordinates into the Formula

Replace the variables in the equation with the coordinates: \[ d = \sqrt{(2.3 - (-0.2))^{2} + (1.1 - 0.3)^{2}} \].

Simplify Inside the Parentheses

Calculate the values inside the parentheses: \[ 2.3 - (-0.2) = 2.3 + 0.2 = 2.5 \] and \[ 1.1 - 0.3 = 0.8 \]. The formula now looks like \[ d = \sqrt{(2.5)^{2} + (0.8)^{2}} \].

Square the Differences

Calculate the squares of the differences: \[ (2.5)^{2} = 6.25 \] and \[ (0.8)^{2} = 0.64 \]. The formula now is \[ d = \sqrt{6.25 + 0.64} \].

Sum the Squares

Add the squared differences: \[ 6.25 + 0.64 = 6.89 \]. Hence, the distance formula now looks like \[ d = \sqrt{6.89} \].

Find the Square Root

Calculate the square root of 6.89: \[ d = \sqrt{6.89} \approx 2.626 \].

Key concepts

coordinate plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It has two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin, denoted as (0,0). By using these axes, we can define any point in the plane with a pair of numbers called coordinates.

For instance, in the problem, we have two points: \(P_{1} = (-0.2, 0.3)\) and \(P_{2} = (2.3, 1.1)\). Here, the first number in each pair represents the x-coordinate, and the second number represents the y-coordinate.

Understanding how to plot and interpret these points on a coordinate plane is crucial for solving problems like finding the distance between two points.

distance between points

Calculating the distance between two points on a coordinate plane is a common task in geometry and algebra. To find this distance, we generally use the distance formula, which is \[ d = \sqrt{ (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} } \].

This formula might look intimidating at first, but it essentially finds the length of the line connecting two points. It uses the differences in the x-coordinates and y-coordinates of the points to do so. Let's break it down:

• \(x_{1}\) and \(y_{1}\) are the coordinates of the first point.
• \(x_{2}\) and \(y_{2}\) are the coordinates of the second point.

By substituting these values into the formula, we can find the distance. For example, using our points \(P_{1}\) and \(P_{2}\), we get \[d = \sqrt{ (2.3 - (-0.2))^{2} + (1.1 - 0.3)^{2} } \], which simplifies to \[d = \sqrt{ (2.5)^{2} + (0.8)^{2} } \].

Lastly, find the square root of the sum of these squares to get the final distance. Understanding this step-by-step process is key to mastering the distance formula.

Pythagorean theorem

The distance formula is derived from the Pythagorean theorem, which is a fundamental principle in geometry. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it is expressed as: \(a^{2} + b^{2} = c^{2}\).

In the context of the coordinate plane, the distance between two points represents the hypotenuse of a right triangle where the differences in the x and y coordinates are the other two sides. For our points \(P_{1}\) and \(P_{2}\), we can visualize a right triangle where:

• The vertical leg has a length of \(1.1 - 0.3 = 0.8\).
• The horizontal leg has a length of \(2.3 - (-0.2) = 2.5\).

Applying the Pythagorean theorem:

• The horizontal leg squared: \(2.5^{2} = 6.25\).
• The vertical leg squared: \(0.8^{2} = 0.64\).

Adding these gives us the square of the hypotenuse: \(6.25 + 0.64 = 6.89\).

Finally, we take the square root of 6.89 to find the length of the hypotenuse, which is the distance between our two points: \(\sqrt{6.89} ≈ 2.626\). Integrating the Pythagorean theorem helps solidify the understanding of the distance formula.