Distance Formula

The Organic Chemistry Tutor
29 Dec 201706:32
EducationalLearning
32 Likes 10 Comments

TLDRThis educational video script offers a clear explanation of how to use the distance formula to calculate the distance between two points. It demonstrates the process using two examples, with points A(1,2) and B(9,17), and points C(5,-16) and D(-2,8), showing the steps to find the distance between them. Additionally, the script teaches how to calculate the area of a circle by determining the radius using the distance formula and then applying the formula for the area of a circle, Ο€rΒ². The example given uses the center of the circle at (2,1) and point P(6,4) to find the radius and subsequently the area, resulting in 25Ο€ square units.

Takeaways
  • πŸ“ The video demonstrates how to use the distance formula to find the distance between two points.
  • πŸ“ The coordinates of two points are essential to apply the distance formula, with the example points A (1,2) and B (9,17).
  • πŸ”’ The distance formula is \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), which requires subtraction of coordinates and squaring the results.
  • πŸ“‰ For points A and B, the calculation involves \(9 - 1 = 8\) for the x-difference and \(17 - 2 = 15\) for the y-difference, leading to \(8^2 + 15^2 = 289\).
  • πŸ›‘ The square root of 289 is 17, so the distance between points A and B is 17 units.
  • πŸ“ The video provides a second example using points C (5, -16) and D (-2, 8) to practice the distance formula.
  • πŸ” The calculation for points C and D involves negative numbers, resulting in a distance of 25 units.
  • 🌐 The video also covers calculating the area of a circle given the center and a point on the circle.
  • πŸ“ˆ The radius of the circle is found using the distance formula between the center (2,1) and point P (6,4), which yields a radius of 5 units.
  • πŸ“Š The area of a circle is calculated using the formula \(\pi r^2\), resulting in \(25\pi\) square units for the given example.
  • πŸŽ“ The video serves as an educational resource for understanding the application of the distance formula and circle area calculations.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is how to use the distance formula to calculate the distance between two points.

  • What are the coordinates of Point A mentioned in the video?

    -The coordinates of Point A are (1, 2).

  • What are the coordinates of Point B mentioned in the video?

    -The coordinates of Point B are (9, 17).

  • What is the distance formula used to calculate the distance between two points?

    -The distance formula is \( \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \).

  • What is the calculated distance between Point A and Point B?

    -The calculated distance between Point A and Point B is 17 units.

  • What are the coordinates of Point C mentioned in the video?

    -The coordinates of Point C are (5, -16).

  • What are the coordinates of Point D mentioned in the video?

    -The coordinates of Point D are (-2, 8).

  • What is the calculated distance between Point C and Point D?

    -The calculated distance between Point C and Point D is 25 units.

  • How is the distance formula applied to calculate the radius of a circle?

    -The distance formula is applied by calculating the distance between the center of the circle and a point on the circle, which gives the radius.

  • What is the formula to calculate the area of a circle?

    -The formula to calculate the area of a circle is \( \pi r^2 \), where \( r \) is the radius.

  • What is the calculated area of the circle with center at (2, 1) and point P at (6, 4)?

    -The calculated area of the circle is \( 25\pi \) square units.

Outlines
00:00
πŸ“ Distance Formula Application

This paragraph introduces the concept of using the distance formula to calculate the distance between two points in a coordinate plane. It provides a step-by-step example using points A(1,2) and B(9,17), explaining how to apply the formula \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) to find the distance, which is 17 units. The paragraph also includes a second example with points C(5,-16) and D(-2,8), demonstrating the calculation and resulting in a distance of 25 units. The explanation is clear and focuses on understanding the components of the distance formula and how to apply them.

05:01
πŸ“ Calculating the Area of a Circle Using Radius

This paragraph demonstrates how to calculate the area of a circle when given the center and a point on the circumference. It uses the center of the circle at (2,1) and point P at (6,4) to illustrate the process. The distance formula is applied to find the radius of the circle, which is determined to be 5 units by calculating the distance between the center and point P. The area of the circle is then found using the formula \(\pi r^2\), resulting in \(25\pi\) square units. The explanation emphasizes the importance of finding the radius to calculate the area and provides a clear method for doing so.

Mindmap
Keywords
πŸ’‘Distance Formula
The Distance Formula is a mathematical formula used to calculate the distance between two points in a Cartesian coordinate system. It is essential in this video as it serves as the primary method for finding the distance between points A and B, and points C and D. The formula is defined as the square root of (x2 - x1) squared plus (y2 - y1) squared, where (x1, y1) and (x2, y2) are the coordinates of the two points. In the script, it is applied twice to find the distances between different pairs of points.
πŸ’‘Coordinates
Coordinates are pairs of numerical values that determine the position of a point in a two-dimensional space. In the context of the video, coordinates are used to represent the locations of points A, B, C, and D. The script provides the coordinates of these points as (1, 2) for point A, (9, 17) for point B, (5, -16) for point C, and (-2, 8) for point D, which are then used in the Distance Formula to calculate distances.
πŸ’‘Square Root
A square root is a value that, when multiplied by itself, gives the original number. In the video, the square root is used in the Distance Formula to find the actual distance between two points after squaring the differences in their coordinates. For example, after calculating the sum of squares of differences in the coordinates of points A and B, the square root of the result (289) is taken to find the distance of 17 units.
πŸ’‘Squaring
Squaring is the mathematical operation of multiplying a number by itself. In the video, squaring is used in the Distance Formula to calculate the differences between the x and y coordinates of two points. For instance, the script mentions squaring the difference between x1 and x2 (8 squared is 64) and y1 and y2 (15 squared is 225) to apply the formula correctly.
πŸ’‘Radius
The radius of a circle is the distance from its center to any point on its circumference. In the video, the radius is determined by calculating the distance between the center of the circle (2, 1) and a point on the circle (6, 4) using the Distance Formula. This radius is then used to calculate the area of the circle.
πŸ’‘Area
Area refers to the amount of space enclosed within a two-dimensional shape. In the context of the video, the area is used to describe the size of a circle. The script explains that the area of a circle can be calculated using the formula pi times the radius squared (Ο€r^2). After finding the radius to be 5 units, the area is calculated as 25Ο€ square units.
πŸ’‘Circle
A circle is a two-dimensional shape where all points are equidistant from a central point known as the center. In the video, a circle is described with its center at coordinates (2, 1) and a point on the circle at (6, 4). The distance between the center and the point is used to determine the radius, which is essential for calculating the circle's area.
πŸ’‘Graph
A graph is a visual representation used to display data or mathematical functions. In the video, a graph is mentioned as a tool to visualize the circle and the points involved. It helps to illustrate the spatial relationship between the center of the circle and point P, which is essential for understanding how the radius is determined.
πŸ’‘Practice
Practice refers to the act of performing an activity repeatedly to improve or master it. In the video, the script suggests practicing the use of the Distance Formula by applying it to different sets of points, such as points C and D. This practice is intended to reinforce the understanding and application of the formula.
πŸ’‘Pi (Ο€)
Pi, represented by the symbol Ο€, is a mathematical constant that represents the ratio of a circle's circumference to its diameter. In the video, pi is used in the formula for calculating the area of a circle (Ο€r^2). The script mentions that after finding the radius, pi is multiplied by the square of the radius to find the area of the circle.
Highlights

Introduction to using the distance formula to calculate the distance between two points.

Example calculation using points A (1,2) and B (9,17) with the distance formula.

Explanation of the distance formula components: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step-by-step calculation showing \(9 - 1 = 8\) and \(17 - 2 = 15\).

Demonstration of squaring the differences: \(8^2 = 64\) and \(15^2 = 225\).

Summation of squared differences: \(64 + 225 = 289\).

Final calculation of the square root of 289 to find the distance is 17 units.

Practice example with points C (5, -16) and D (-2, 8).

Calculation steps for the practice example using the distance formula.

Explanation of the calculation: \((-2 - 5) = -7\) and \(8 - (-16) = 24\).

Squaring the results: \((-7)^2 = 49\) and \(24^2 = 576\).

Summation and square root to find the distance between C and D is 25 units.

Introduction to calculating the area of a circle using a point on the circle and its center.

Graphical representation of the circle with center at (2,1) and point P at (6,4).

Use of the distance formula to find the radius of the circle.

Calculation steps for the radius using the coordinates of the center and point P.

Finding the radius to be 5 units using the distance formula.

Final calculation of the area of the circle using the formula \(\pi r^2\) resulting in \(25\pi\) square units.

Transcripts

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