Distance Formula | Introduction to Analytic Geometry|

Summary

This video tutorial introduces analytic geometry, focusing on the distance formula within the Cartesian coordinate system. It explains how to calculate the distance between two points using coordinates (x1, y1) and (x2, y2), applying the Pythagorean theorem. The formula d = √[(x2 - x1)² + (y2 - y1)²] is derived and demonstrated with sample problems, including a practical example and a reverse engineering approach to find the value of y when given a point and distance.

Takeaways

• 📚 Analytic geometry is a branch of mathematics that utilizes rectangular coordinates, also known as the Cartesian coordinate system.
• 🔍 The Cartesian coordinate system is named after René Descartes, who is considered the founder of analytic geometry.
• 📐 The distance formula is a fundamental concept in analytic geometry, used to calculate the distance between two points in a plane.
• 📈 The Cartesian coordinate plane consists of four quadrants, with the x-axis and y-axis being the intersecting lines that define these quadrants.
• 📍 A point in the coordinate plane is represented by its x and y coordinates, such as point P(x, y).
• 📝 The distance formula is derived from the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
• 🧩 The formula for the distance (d) between two points (x1, y1) and (x2, y2) is given by \( \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \).
• 🔢 The script provides a sample problem using the distance formula to calculate the distance between two points with given coordinates.
• 🔄 The script also discusses a reverse engineering approach to solving problems where the distance is given, and one needs to find the coordinates.
• 📉 The transcript includes a step-by-step guide on how to use a calculator to compute the distance between two points using the formula.
• 🔑 The final takeaway is a problem-solving example where the distance formula is used to find the value of 'y' when the coordinates of two points and the distance between them are known.

Q & A

What is analytic geometry?

Analytic geometry is a branch of mathematics that uses a coordinate system to define and analyze geometric objects. It is also known as coordinate geometry or Cartesian geometry, named after René Descartes.

What is the Cartesian coordinate plane?

The Cartesian coordinate plane is a two-dimensional plane that is defined by two perpendicular axes: the horizontal x-axis and the vertical y-axis. It is used to graph points and geometric shapes using ordered pairs of numbers.

What are the four quadrants of the Cartesian coordinate plane?

The four quadrants of the Cartesian coordinate plane are numbered counterclockwise starting from the upper right. They are: the first quadrant (+,+), the second quadrant (-,+), the third quadrant (-,-), and the fourth quadrant (+,-).

What is the distance formula used for?

The distance formula is used to calculate the distance between two points in a coordinate plane. It is derived from the Pythagorean theorem and is expressed as \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

How is the distance formula derived?

The distance formula is derived from the Pythagorean theorem by considering the coordinates of two points as the legs of a right triangle and the distance between the points as the hypotenuse.

What are the coordinates of point P mentioned in the script?

The script does not provide specific coordinates for point P, but it describes point P as being represented by its coordinates (x, y) on the Cartesian coordinate plane.

What is the purpose of the sample problem involving points A(-3, -2) and B(5, 5)?

The purpose of the sample problem is to demonstrate how to use the distance formula to calculate the distance between two given points, A and B, in the Cartesian coordinate plane.

How is the distance between points A(-3, -2) and B(5, 5) calculated in the script?

The distance is calculated using the distance formula: \( D = \sqrt{(5 - (-3))^2 + (5 - (-2))^2} \), which simplifies to \( D = \sqrt{8^2 + 7^2} \) and results in a distance of 10 units.

What is the second problem in the script about?

The second problem involves finding the value of y in the point (3, y + 8) when the distance between this point and the point (7, y) is given as 13.

How is the value of y determined in the second problem?

The value of y is determined by applying the distance formula to the given points and solving for y. The script suggests using reverse engineering from the given options to find the correct value of y, which is -5.

What is reverse engineering in the context of the second problem?

Reverse engineering in this context refers to solving the problem by starting from the given distance and working backward to find the unknown variable, y, using the distance formula and the provided options.

Outlines

00:00

📚 Introduction to Analytic Geometry and Distance Formula

This paragraph introduces the concept of analytic geometry, which is a branch of mathematics that utilizes the Cartesian coordinate plane to define and analyze geometric shapes algebraically. The focus is on the distance formula, a fundamental tool used to calculate the distance between two points in the plane. The Cartesian coordinate system is explained, consisting of four quadrants and two perpendicular axes, the x-axis and y-axis. A point is represented by its coordinates (x, y). The distance formula is derived from the Pythagorean theorem and is expressed as the square root of the sum of the squares of the differences in the x and y coordinates of the two points. An example is provided to demonstrate the calculation of the distance between two given points.

05:01

🔢 Applying the Distance Formula with a Calculator

The second paragraph delves into the practical application of the distance formula using a calculator. It emphasizes the importance of setting the calculator to the correct mode for accurate computations. The process involves entering the differences in the x and y coordinates of two points, squaring these differences, summing them, and then taking the square root of the result to find the distance. A sample calculation is presented, demonstrating how to input and compute the values to find the distance between two points with specific coordinates. Additionally, the paragraph touches on a problem-solving scenario where the distance between two points is given, and one of the coordinates is unknown, requiring the use of the distance formula in reverse to solve for the unknown value.

Keywords

💡Analytic Geometry

Analytic Geometry is a branch of mathematics that uses a coordinate system to define and analyze geometric objects. In the video, it is the main topic being discussed, with an emphasis on using the Cartesian coordinate plane for calculations. The script introduces analytic geometry as a geometry that employs rectangular coordinates, highlighting its foundational role in the tutorial.

💡Cartesian Coordinate Plane

The Cartesian Coordinate Plane is a two-dimensional plane where points are defined by their x and y coordinates, named after René Descartes. In the video, it is the setting where geometric concepts are translated into algebraic equations, and it is described as having four quadrants and two perpendicular axes, the x-axis and y-axis.

💡Distance Formula

The Distance Formula is a mathematical formula used to calculate the distance between two points in a coordinate plane. The script explains the formula as the square root of the sum of the squares of the differences in the x-coordinates and y-coordinates of the two points, which is central to the video's demonstration of analytic geometry.

💡Quadrants

Quadrants refer to the four equal areas formed by the intersection of the x-axis and y-axis in the Cartesian Coordinate Plane. The script mentions the first, second, third, and fourth quadrants, which are essential for understanding the location of points and their coordinates.

💡Axis

In the context of the Cartesian Coordinate Plane, an Axis is a straight line that serves as a reference for the coordinates of points. The script describes the x-axis as the horizontal line and the y-axis as the vertical line, which intersect at the origin (0,0).

💡Coordinates

Coordinates are the numerical values that define a point's location in a coordinate system. The script uses the term to describe how a point P is represented by its x and y values, which is fundamental to understanding the position of points in analytic geometry.

💡Pythagorean Theorem

The Pythagorean Theorem is a principle in geometry that states the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. The script uses this theorem to derive the distance formula, showing its application in analytic geometry.

💡Hypotenuse

The Hypotenuse is the longest side of a right-angled triangle, opposite the right angle. In the script, the hypotenuse is the line representing the distance between two points, and its length is calculated using the distance formula derived from the Pythagorean Theorem.

💡Sample Problem

A Sample Problem in the script is an example used to illustrate the application of the distance formula. It involves calculating the distance between two given points, demonstrating how the formula is applied in practice within the context of analytic geometry.

💡Reverse Engineering

Reverse Engineering in the script refers to the process of working backward from a known result to find the unknown variables. This is shown in the second sample problem, where the distance between points is given, and the value of y is to be determined using the distance formula.

💡Calculator

A Calculator is a device or software used to perform mathematical calculations. The script mentions the use of a calculator to compute the distance between two points, emphasizing the practical application of the distance formula and the importance of computational tools in analytic geometry.

Highlights

Introduction to analytic geometry, a branch of mathematics that utilizes rectangular coordinates.

The Cartesian coordinate plane, named after René Descartes, is the foundation of analytic geometry.

Explanation of the four quadrants of the Cartesian coordinate plane.

Description of the x-axis and y-axis as the intersecting lines of the coordinate plane.

Introduction to the distance formula as a method to calculate the distance between two points.

Illustration of the distance formula using a right triangle formed by two points.

The Pythagorean theorem is applied to derive the distance formula.

The mathematical expression for the distance formula is presented.

A sample problem is introduced to demonstrate the application of the distance formula.

Calculation of the distance between two given points in the Cartesian plane.

Use of a calculator to find the numerical distance between points.

Introduction to a problem involving finding the value of 'y' given the distance between two points.

The concept of reverse engineering in solving geometry problems is explained.

A step-by-step guide on how to use the distance formula to solve for an unknown variable.

Demonstration of the process to eliminate incorrect answers and find the correct solution.

Conclusion that the correct value of 'y' satisfies the condition of the given distance.

Transcripts

hi guys in this video we will start our

discussion in analytic geometry so when

we say analytic geometry it is a

geometry that uses rectangular

coordinates or unilateral not in a

popularly known as the Cartesian

coordinate plane

so encourage a c2 or any de cartes the

founder of analytic geometry so in this

first part of our video tutorial we will

talk about the distance formula when you

say it distance formula it is used to

calculate the distance or the distance

between two points so let's take a quick

look in our Cartesian coordinate things

or a Cartesian coordinate plane is

composed of four quadrants you have this

we have the first quadrant you have the

first the second quadrant third and the

fourth quadrant and these two

intersecting lines which are

perpendicular to each other is what you

call the axis so this horizontal is the

x axis and the vertical is the y axis

then when you put a point let's say say

this is a point P say P the point P is

described by its coordinates x and y now

let's proceed to our work distance

formula so some googling young distance

formula suppose that we have two points

a point one so this point one is you

have x1 y1 and point

so you have x2 y2 so the distance

formula is used to compute this distance

so the distance between two points so

this is B now how do you can get that

the concept is using your pedegg

orienteer M so if we extend this point

here in this point here thus forming a

right triangle you have this point here

so this point is you have X sub 2 then Y

sub 1 then this distance line here is

given by you have Y sub 2 course your

boy sub 2 minus y sub 1 the height in

this distance is X sub 2 minus X sub 1

so using the Pythagorean theorem you

have the square of the hypotenuse is

equal to the square or the sum of the

squares of the two side so you have d

squared equals you have take sub 2 minus

X sub 1 squared plus y sub 2 minus y sub

1 squared so the distance is equal to

you have the square root of x sub 2

minus X sub 1 squared plus y sub 2 minus

y sub 1 squared so this is our distance

formula now that's so our first sample

problem

you

so we have these two points for a

negative 3 negative 2 5 so we are

required to find the distance between

those two points so you have a that is 4

so NASA fourth quadrant shop so for 10

decorative 3 so this is 8 point a is 4

negative 3 when point B is in the second

quadrant so negative 2 5 you have

negative 2 5

so the distance is this length so this

length distance we can so for that using

the distance formula here D equals the

square root of we have X sub 2 minus X

sub 1 squared plus y sub 2 minus y sub 1

squared P 2d equals we have xn 2 you

have negative 2 minus X sub 1 so 4

squared plus y sub 2 5 minus negative 3

squared so D equals

we have been unit so the distance is 10

units so that allows me room publish on

calculator technique or sit down by base

mode so be sure none calculator is naka

mode you have computations of 1 then

let's proceed to your shift then pull

young plus so goombahs noon put in

entered and nothing no difference no

Hmong xn twice sample here you have 4

minus negative 2 so 4 minus negative 2

then comma so ship so my is negative 3

so negative 3 minus 5 then parentheses

London equals so my get nothing done

guys young 10 is your distance

so when nothing a meeting in contacting

you formula

so in this next problem here if the

distance between points 3y + 8 7 is 13

then y is equal to so we given time and

distance so you need not not and stock

already need so in this problem feed

emotion compute using a distance formula

but in peanut remodeling solution if

you're given the choice as Esther

bollocks on guys you're given the

choices you can start working from the

option or choices to the problem it's

what you call the reverse engineering so

pundit I didn't get a car tech nothing

so first and nothing moved up but now

computation tires of one then polar

audit to shift pull

so plus then you have three minus eight

you have 3 minus 8 then coma

oh my parentheses then so why -

I've been so Alpha X minus 7 then occult

not injure so coxy and see Yuma

partnership then input nothing as you

involve unum

NASA choice SI 5 so bug in equals not in

the opportune are is 13 so if not say

the 5.35 it means that

five is not the correct answer it means

that this is removed from the auction so

militia automatic then stay nothing you

later see that that is 19 so cut so 19

equals to 13th of correction so possible

answer so boom oh nothing Sabine and see

another guy see you later this parenting

Shang 19 so it right not in you negative

five so call you have negative 5 so

negative 5 we have 13 also treatments

that letter D is our answer since the

negative 5 and 19 satisfies the

condition that the distance is 13