Question

Find the distance \(d\) between the points \(P_{1}\) and \(P_{2}\). $$ P_{1}=(-4,-3) ; \quad P_{2}=(6,2) $$

Short answer

The distance between the points is \ 5\sqrt{5}.

Step-by-step solution

- Write Down the Distance Formula

The distance formula between two points \(x_1, y_1\) and \(x_2, y_2\) in a coordinate plane is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \(d\) is the distance between the two points.

- Substitute the Coordinates into the Formula

Plug in the coordinates of \(P_1 = (-4, -3)\) and \(P_2 = (6, 2)\) into the distance formula: \[ d = \sqrt{(6 - (-4))^2 + (2 - (-3))^2} \]

- Simplify Inside the Parentheses

Simplify the expressions inside the parentheses: \[ d = \sqrt{(6 + 4)^2 + (2 + 3)^2} = \sqrt{10^2 + 5^2} \]

- Square the Differences

Calculate the squares of the differences: \[ d = \sqrt{100 + 25} \]

- Add the Results

Add the squared differences: \[ d = \sqrt{125} \]

- Take the Square Root

Finally, take the square root of 125 to find the distance: \[ d = \sqrt{125} = 5\sqrt{5} \]

Key concepts

coordinate plane

In geometry, we use a plane to plot points and shapes. This plane is called the coordinate plane, or Cartesian plane. It is divided into four quadrants by two perpendicular lines:

The horizontal line is called the x-axis.

• The vertical line is called the y-axis.

Points on the coordinate plane are identified by ordered pairs \((x, y)\).
Here is a brief breakdown:
- The first number in the pair, x, refers to the horizontal position.
- The second number, y, refers to the vertical position.

Being able to identify and plot points on the coordinate plane is fundamental to understanding distance between points.

distance between points

The distance formula helps us to find the length between two points on the coordinate plane. This formula is very useful in geometry and many other fields. Here's how it works:

Given two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) between them is found with:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

In plain words, the distance formula applies the Pythagorean theorem to find the length of the hypotenuse of a right triangle formed by the points. This involves:

• Finding the difference in x-coordinates between the two points and squaring it.
• Finding the difference in y-coordinates and squaring that.
• Adding these squares together.
• Taking the square root of the sum to get the final distance.

square root

The square root is a very important mathematical function, especially when working with distances. The square root of a number \(n\), denoted as \(\sqrt{n}\), is a value that, when multiplied by itself, equals \(n\). For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

In the distance formula, the square root is used after summing up the squared differences to find the exact distance. This step is crucial:

• It converts the squared units back into the original units of measurement.
• It provides an accurate numerical value for the distance.
• Without taking the square root, we would get the sum of squared differences, not the actual distance.

Understanding how to handle square roots is essential for solving many geometric problems, including finding the distance between two points on a coordinate plane.