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Chapter 2: Problem 25

Find the distance \(d\) between the points \(P_{1}\) and \(P_{2}\). $$ P_{1}=(-7,3) ; \quad P_{2}=(4,0) $$

Short Answer

Expert verified
\( d = \sqrt{130} \)

Step by step solution

01

- Identify Coordinates

Label the coordinates of the points. Let the coordinates of point \(P_1\) be \( (-7, 3) \) and the coordinates of point \(P_2\) be \( (4, 0) \).
02

- Use the Distance Formula

The distance formula in a 2-dimensional plane is given by \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].
03

- Substitute the Coordinates

Substitute the coordinates \((x_1, y_1) = (-7, 3)\) and \((x_2, y_2) = (4, 0)\) into the distance formula: \[ d = \sqrt{(4 - (-7))^2 + (0 - 3)^2} \].
04

- Simplify the Equation

Simplify the expression inside the square root: \[ d = \sqrt{(4 + 7)^2 + (-3)^2} \], which further simplifies to \[ d = \sqrt{11^2 + (-3)^2} \].
05

- Calculate Squared Values

Calculate the squares: \[ d = \sqrt{121 + 9} \].
06

- Add and Find the Square Root

Add the squared values and find the square root: \[ d = \sqrt{130} \]. Finally, simplify \( \sqrt{130} \) (if necessary, leave it as an approximation or in simplest form).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinates
In geometry, points on a plane are represented using coordinates. These coordinates tell us the position of a point along the x (horizontal) and y (vertical) axes. For instance, in the given exercise, we have two points, \(P_1 = (-7, 3)\) and \(P_2 = (4, 0)\). Here, -7 is the x-coordinate of \(P_1\) and 3 is the y-coordinate; similarly, 4 is the x-coordinate of \(P_2\) and 0 is the y-coordinate. Understanding coordinates is fundamental to navigating points on a plane and calculating distances between them.
Euclidean Distance
The Euclidean distance is a measure of the straight-line distance between two points in a plane. It is derived from the Pythagorean theorem and helps us quantify how far apart two points are. To compute the Euclidean distance, we use the distance formula:

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

For our exercise, this means substituting the coordinates \(P_1 = (-7, 3)\) and \(P_2 = (4, 0)\) into the formula:

\[ d = \sqrt{(4 - (-7))^2 + (0 - 3)^2} \]

Breaking it down:
  • Subtract the x-coordinates: \(4 - (-7) = 11 \)
  • Subtract the y-coordinates: \(0 - 3 = -3 \)
  • Square these differences: \(11^2 = 121 \) and \(-3^2 = 9\)
  • Add the squares: \(121 + 9 = 130 \)
  • Find the square root: \( \sqrt{130}\)

Therefore, \( d = \sqrt{130} \) represents the Euclidean distance between the points.
Pythagorean Theorem
The Pythagorean theorem is a fundamental principle in geometry used to find the lengths of sides in a right-angled triangle. According to the theorem:

\[ a^2 + b^2 = c^2 \]

Here, \(a\) and \(b\) are the lengths of the perpendicular sides, and \(c\) is the length of the hypotenuse. In the context of finding the distance between two points, we can visualize a right-angled triangle where the difference between the x-coordinates and y-coordinates forms the perpendicular sides. Using the Pythagorean theorem, we find the hypotenuse (the distance we want):

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