Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 45E
Related questions
Question
![**Question 3: Calculating Distance Between Two Points**
**Problem Statement:**
What is the distance between points \((0, -1)\) and \((10, -5)\)?
**Explanation and Solution:**
To determine the distance between two points in a Cartesian plane, we utilize the Distance Formula, which is derived from the Pythagorean Theorem. The Distance Formula is:
\[
d = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}}
\]
Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
1. Identify the coordinates of the points:
- Point 1: \((x_1, y_1) = (0, -1)\)
- Point 2: \((x_2, y_2) = (10, -5)\)
2. Plug the coordinates into the Distance Formula:
\[
d = \sqrt{{(10 - 0)^2 + (-5 - (-1))^2}}
\]
3. Simplify inside the square root:
\[
d = \sqrt{{10^2 + (-5 + 1)^2}}
\]
\[
d = \sqrt{{10^2 + (-4)^2}}
\]
\[
d = \sqrt{{100 + 16}}
\]
\[
d = \sqrt{{116}}
\]
4. Simplify the square root (if possible):
\[
d = 2\sqrt{29}
\]
**Conclusion:**
The distance between the points \((0, -1)\) and \((10, -5)\) is \(2\sqrt{29}\) units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3829f931-9257-40ee-99af-de1dd428439e%2F2e706d72-f10f-4166-b6e7-176b4cc5ae9e%2Findgp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 3: Calculating Distance Between Two Points**
**Problem Statement:**
What is the distance between points \((0, -1)\) and \((10, -5)\)?
**Explanation and Solution:**
To determine the distance between two points in a Cartesian plane, we utilize the Distance Formula, which is derived from the Pythagorean Theorem. The Distance Formula is:
\[
d = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}}
\]
Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
1. Identify the coordinates of the points:
- Point 1: \((x_1, y_1) = (0, -1)\)
- Point 2: \((x_2, y_2) = (10, -5)\)
2. Plug the coordinates into the Distance Formula:
\[
d = \sqrt{{(10 - 0)^2 + (-5 - (-1))^2}}
\]
3. Simplify inside the square root:
\[
d = \sqrt{{10^2 + (-5 + 1)^2}}
\]
\[
d = \sqrt{{10^2 + (-4)^2}}
\]
\[
d = \sqrt{{100 + 16}}
\]
\[
d = \sqrt{{116}}
\]
4. Simplify the square root (if possible):
\[
d = 2\sqrt{29}
\]
**Conclusion:**
The distance between the points \((0, -1)\) and \((10, -5)\) is \(2\sqrt{29}\) units.
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