Materie

Matematica

Geometria

Coniche e trasformazioni

Scopri il Piano Cartesiano e Il Baricentro Divertendoti!

Name: Knowunity
Brand: Knowunity

Vedi

Scopri il Piano Cartesiano e Il Baricentro Divertendoti!

Valentina Grecucci

@valentinagrecuccii

912 Follower

Segui

Il piano cartesiano è uno strumento fondamentale per rappresentare punti e figure geometriche in matematica. Consiste in due assi perpendicolari che si intersecano nell'origine, dividendo il piano in quattro quadranti. Ogni punto è identificato da una coppia di coordinate (x,y). Il piano cartesiano permette di calcolare distanze tra punti, trovare punti medi di segmenti e determinare il baricentro di triangoli.

• Il piano cartesiano è composto da asse x (ascisse) e asse y (ordinate)
• Le coordinate (x,y) identificano univocamente ogni punto del piano
• Si possono calcolare distanze tra punti e trovare punti medi di segmenti
• Il baricentro di un triangolo si trova all'intersezione delle mediane
• Formule specifiche permettono calcoli precisi su punti e figure geometriche

4/11/2022

2017

Piano cartesiano
ASSI cartesiani
13/10/2022
Fissare un sistema di assi cartesiani, cioè disegnare un piano cartesiano
Asse delle y o asse de

Introduction to the Cartesian Plane

The Cartesian plane, also known as the coordinate plane, is a fundamental concept in mathematics that allows us to represent points and geometric figures in two-dimensional space. This page introduces the key components and principles of the Cartesian plane.

The Cartesian plane consists of two perpendicular number lines called axes:

The horizontal axis is called the x-axis or axis of abscissas
The vertical axis is called the y-axis or axis of ordinates

The point where these axes intersect is called the origin, denoted as O, with coordinates (0,0).

Every point on the plane can be represented by an ordered pair of numbers (x, y), where:

x is the distance from the y-axis (abscissa)
y is the distance from the x-axis (ordinate)

The plane is divided into four quadrants, numbered counterclockwise from the top right:

First quadrant (+,+): Both x and y are positive
Second quadrant (-,+): x is negative, y is positive
Third quadrant (-,-): Both x and y are negative
Fourth quadrant (+,-): x is positive, y is negative

Vocabulary: Abscissa - The x-coordinate of a point Vocabulary: Ordinate - The y-coordinate of a point

Highlight: The arrows on the axes are important as they indicate that values increase as you move away from the origin.

Example: Point A (-2, 3) is in the second quadrant, while point B (4, 0) is on the positive x-axis.

Vedi

Midpoint and Centroid

This page covers two important concepts in coordinate geometry: the midpoint of a line segment and the centroid of a triangle.

Punto medio di un segmento formula (Midpoint Formula): For a line segment with endpoints A(xA, yA) and B(xB, yB), the coordinates of the midpoint M are: M = ((xA + xB)/2, (yA + yB)/2)

Example: For a segment with endpoints A(5, -3) and B(1, 6), the midpoint M is: xM = (5 + 1)/2 = 3 yM = (-3 + 6)/2 = 1.5 Therefore, M(3, 1.5)

Baricentro di un triangolo formula (Centroid of a Triangle): The centroid is the point where the three medians of a triangle intersect. A median is a line segment that connects a vertex to the midpoint of the opposite side.

The formula for the centroid G of a triangle with vertices A(x1, y1), B(x2, y2), and C(x3, y3) is: G = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)

Highlight: The centroid divides each median in a 2:1 ratio, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex.

Vocabulary: Median - A line segment that connects a vertex of a triangle to the midpoint of the opposite side.

These formulas are essential for solving various geometric problems and are frequently used in Punto medio esercizi pdf and Baricentro triangolo equilatero formula applications.

Vedi

Distance Between Two Points

This page focuses on calculating the distanza tra due punti (distance between two points) in the Cartesian plane. The method for calculating this distance depends on the relative positions of the points.

When two points have the same ordinate (y-coordinate): d(A,B) = |xB - xA|

When two points have the same abscissa (x-coordinate): d(A,C) = |yC - yA|

For the general case, when points have different x and y coordinates, we use the distance formula: d(B,C) = √[(xC - xB)² + (yC - yB)²]

This formula is derived from the Pythagorean theorem and calculates the length of the hypotenuse of a right triangle formed by the two points and their projections on the axes.

Definition: The distance formula calculates the shortest path between two points in the Cartesian plane.

Example: To find the distance between points B(xB, yB) and C(xC, yC), we use the formula: d(B,C) = √[(xC - xB)² + (yC - yB)²]

Highlight: The distance formula is a fundamental tool in analytic geometry and is used in many applications, including Distanza tra due punti Maps calculations.

Vedi

Recap and Key Concepts

This page provides a comprehensive recap of the key concepts covered in the previous pages, reinforcing the fundamental principles of the Cartesian plane and coordinate geometry.

Points and Segments in the Cartesian Plane:

The Cartesian plane consists of two perpendicular axes: the x-axis (abscissa) and the y-axis (ordinate).
The intersection point of these axes is called the origin O(0,0).
Each point P in the plane is represented by an ordered pair of real numbers (xP, yP).
The plane is divided into four quadrants, numbered counterclockwise from the top right.

Distanza tra due punti: formula:

For points with the same ordinate: d(A,B) = |xB - xA|
For points with the same abscissa: d(A,B) = |yB - yA|
General case: d(A,B) = √[(xB - xA)² + (yB - yA)²]

Punto medio tra due punti:

For a segment AB, the midpoint M is given by: M = ((xA + xB)/2, (yA + yB)/2)

Highlight: Points on the x-axis have y-coordinate 0, while points on the y-axis have x-coordinate 0.

Example: To find the distance between A(3,7) and B(9,-1): AB = √[(9-3)² + (-1-7)²] = √[36 + 64] = √100 = 10

These concepts are fundamental for solving various geometric problems and are frequently used in Distanza tra due punti piano cartesiano esercizi and Punto medio di un segmento esercizi svolti.

Vedi

Contenuti simili

Know Formulario Piano Cartesiano e Retta thumbnail

2319

3ªl

Formulario Piano Cartesiano e Retta

Formulario delle principali regole e formule del piano cartesiano e dell'equazione della retta.

104

2285

3ªl

Geometria analitica 1

Breve schema di geometria analitica

Know formulario retta e coniche thumbnail

834

12124

3ªl

formulario retta e coniche

formulario con le principali proprietà e formule di retta, circonferenza, parabola, ellisse e iperbole.

Know La retta : teoria ed esercizi thumbnail

729

3ªl

La retta : teoria ed esercizi

Schema completo di geometria analitica realtivo alla retta nel piano cartesiano con esercizi.

2745

3ªl/4ªl

LA PARABOLA : formulr

Formule sulla parabola, rette tangenti, area di un segmento, e fasci di parabola

Know Simulazione Verifica sulla Parabola thumbnail

283

3ªl

Simulazione Verifica sulla Parabola

simulazione verifica parabola

Non c'è niente di adatto? Esplorare altre aree tematiche.

Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

Knowunity è stata inserita in un articolo di Apple ed è costantemente in cima alle classifiche degli app store nella categoria istruzione in Germania, Italia, Polonia, Svizzera e Regno Unito. Unisciti a Knowunity oggi stesso e aiuta milioni di studenti in tutto il mondo.

Scarica

Google Play

Scarica

App Store

Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

4.9+

Valutazione media dell'app

13 M

Studenti che usano Knowunity

#1

Nelle classifiche delle app per l'istruzione in 12 Paesi

950 K+

Studenti che hanno caricato appunti

Non siete ancora sicuri? Guarda cosa dicono gli altri studenti...

Utente iOS

Adoro questa applicazione [...] consiglio Knowunity a tutti!!! Sono passato da un 5 a una 8 con questa app

Stefano S, utente iOS

L'applicazione è molto semplice e ben progettata. Finora ho sempre trovato quello che stavo cercando

Susanna, utente iOS

Adoro questa app ❤️, la uso praticamente sempre quando studio.

Materie

Matematica

Geometria

Coniche e trasformazioni

Scopri il Piano Cartesiano e Il Baricentro Divertendoti!

Valentina Grecucci

@valentinagrecuccii

912 Follower

Segui

4/11/2022

2017

3ªl

Matematica

Introduction to the Cartesian Plane

The Cartesian plane consists of two perpendicular number lines called axes:

The horizontal axis is called the x-axis or axis of abscissas
The vertical axis is called the y-axis or axis of ordinates

The point where these axes intersect is called the origin, denoted as O, with coordinates (0,0).

Every point on the plane can be represented by an ordered pair of numbers (x, y), where:

x is the distance from the y-axis (abscissa)
y is the distance from the x-axis (ordinate)

The plane is divided into four quadrants, numbered counterclockwise from the top right:

First quadrant (+,+): Both x and y are positive
Second quadrant (-,+): x is negative, y is positive
Third quadrant (-,-): Both x and y are negative
Fourth quadrant (+,-): x is positive, y is negative

Vocabulary: Abscissa - The x-coordinate of a point Vocabulary: Ordinate - The y-coordinate of a point

Highlight: The arrows on the axes are important as they indicate that values increase as you move away from the origin.

Example: Point A (-2, 3) is in the second quadrant, while point B (4, 0) is on the positive x-axis.

Midpoint and Centroid

This page covers two important concepts in coordinate geometry: the midpoint of a line segment and the centroid of a triangle.

Punto medio di un segmento formula (Midpoint Formula): For a line segment with endpoints A(xA, yA) and B(xB, yB), the coordinates of the midpoint M are: M = ((xA + xB)/2, (yA + yB)/2)

Example: For a segment with endpoints A(5, -3) and B(1, 6), the midpoint M is: xM = (5 + 1)/2 = 3 yM = (-3 + 6)/2 = 1.5 Therefore, M(3, 1.5)

The formula for the centroid G of a triangle with vertices A(x1, y1), B(x2, y2), and C(x3, y3) is: G = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)

Highlight: The centroid divides each median in a 2:1 ratio, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex.

Vocabulary: Median - A line segment that connects a vertex of a triangle to the midpoint of the opposite side.

These formulas are essential for solving various geometric problems and are frequently used in Punto medio esercizi pdf and Baricentro triangolo equilatero formula applications.

Distance Between Two Points

When two points have the same ordinate (y-coordinate): d(A,B) = |xB - xA|

When two points have the same abscissa (x-coordinate): d(A,C) = |yC - yA|

For the general case, when points have different x and y coordinates, we use the distance formula: d(B,C) = √[(xC - xB)² + (yC - yB)²]

This formula is derived from the Pythagorean theorem and calculates the length of the hypotenuse of a right triangle formed by the two points and their projections on the axes.

Definition: The distance formula calculates the shortest path between two points in the Cartesian plane.

Example: To find the distance between points B(xB, yB) and C(xC, yC), we use the formula: d(B,C) = √[(xC - xB)² + (yC - yB)²]

Highlight: The distance formula is a fundamental tool in analytic geometry and is used in many applications, including Distanza tra due punti Maps calculations.

Recap and Key Concepts

This page provides a comprehensive recap of the key concepts covered in the previous pages, reinforcing the fundamental principles of the Cartesian plane and coordinate geometry.

Points and Segments in the Cartesian Plane:

The Cartesian plane consists of two perpendicular axes: the x-axis (abscissa) and the y-axis (ordinate).
The intersection point of these axes is called the origin O(0,0).
Each point P in the plane is represented by an ordered pair of real numbers (xP, yP).
The plane is divided into four quadrants, numbered counterclockwise from the top right.