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Impara a Passare dalla Forma Implicita a Esplicita e Calcolare il Punto Medio

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Impara a Passare dalla Forma Implicita a Esplicita e Calcolare il Punto Medio
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rora

@rorasnotes

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164 Follower

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The document explains key concepts in analytic geometry, focusing on linear equations and their representations. It covers converting between implicit and explicit forms of equations, finding midpoints and lengths of line segments, and determining equations of lines given various parameters.

Forma implicita and forma esplicita of linear equations are explained
• Methods for graphing lines and checking if points lie on lines are detailed
• Formulas for midpoint, segment length, and line equations are provided
• Techniques for finding slope (coefficiente angolare) are outlined
• Distance from a point to a line is briefly covered

19/09/2022

1021

ye
y
2 quad. + 1'quad (+;+)
(-;+)
3° quod.
(-;-)
piamo cartesiano
ex:
COME PASSARE DALLA FORMA IMPLICITA A QUELLA ESPLICITA:
+
ax+by+C =0
1.

Vedi

Midpoint Formula and Segment Length

This page focuses on finding the midpoint of a line segment and calculating segment length in the coordinate plane.

The midpoint formula is introduced: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

For segment length, the distance formula is given: AB = √[(x₂ - x₁)² + (y₂ - y₁)²]

Several examples demonstrate the application of these formulas.

The page then transitions to equations of lines, presenting two main methods:

  1. Using two points on the line (point-point form)
  2. Using a point and the slope (point-slope form)

Vocabulary: The punto medio (midpoint) is the point that divides a line segment into two equal parts.

Example: For a segment with endpoints A(2,1) and B(-3,5), the midpoint M is calculated as (-1/2, 3).

Definition: The length of a line segment is the distance between its endpoints, calculated using the distance formula derived from the Pythagorean theorem.

Highlight: The midpoint and length formulas are fundamental tools in coordinate geometry, used in many advanced problems and proofs.

ye
y
2 quad. + 1'quad (+;+)
(-;+)
3° quod.
(-;-)
piamo cartesiano
ex:
COME PASSARE DALLA FORMA IMPLICITA A QUELLA ESPLICITA:
+
ax+by+C =0
1.

Vedi

Slope and Distance to a Line

This page delves deeper into finding the slope (coefficiente angolare) of a line and introduces the concept of distance from a point to a line.

The slope formula is revisited: m = (y₂ - y₁) / (x₂ - x₁)

Examples are provided for calculating slope given two points.

The point-slope form of a line equation is emphasized: y - y₁ = m(x - x₁)

The page concludes with the formula for the distance from a point to a line: d = |ax₀ + by₀ + c| / √(a² + b²)

Where (x₀, y₀) is the point and ax + by + c = 0 is the equation of the line in implicit form.

Definition: The coefficiente angolare (slope) of a line measures its steepness and direction, represented by m in the equation y = mx + q.

Example: For a line passing through A(2,-3) and B(-1,4), the slope is calculated as m = (4 - (-3)) / (-1 - 2) = 7 / -3 = -7/3.

Highlight: Understanding slope is crucial for analyzing the relationship between lines, including parallel and perpendicular lines.

Vocabulary: The distance from a point to a line is the length of the perpendicular segment from the point to the line.

ye
y
2 quad. + 1'quad (+;+)
(-;+)
3° quod.
(-;-)
piamo cartesiano
ex:
COME PASSARE DALLA FORMA IMPLICITA A QUELLA ESPLICITA:
+
ax+by+C =0
1.

Vedi

Converting Between Implicit and Explicit Forms

This page explains how to convert a linear equation from forma implicita to forma esplicita. The process involves isolating the y-variable on one side of the equation.

The general steps are:

  1. Start with the implicit form ax + by + c = 0
  2. Subtract ax and c from both sides
  3. Factor out b from the y term
  4. Divide both sides by b to isolate y

An example is given: 2x + 3y - 6 = 0 is converted to y = -2/3x + 2

The page also covers:

  • Graphing lines using a point-slope approach
  • Checking if a point lies on a line by substituting coordinates
  • Special cases of vertical and horizontal lines

Definition: The forma implicita of a line is ax + by + c = 0, while the forma esplicita is y = mx + q, where m is the slope and q is the y-intercept.

Example: To check if point P(-2,1) lies on y = -3x + 5, substitute the x-coordinate: 1 ?= -3(-2) + 5. Since 1 = 1, the point does lie on the line.

Highlight: Understanding how to convert between implicit and explicit forms is crucial for solving many geometry problems involving lines.

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

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Impara a Passare dalla Forma Implicita a Esplicita e Calcolare il Punto Medio

user profile picture

rora

@rorasnotes

·

164 Follower

Segui

The document explains key concepts in analytic geometry, focusing on linear equations and their representations. It covers converting between implicit and explicit forms of equations, finding midpoints and lengths of line segments, and determining equations of lines given various parameters.

Forma implicita and forma esplicita of linear equations are explained
• Methods for graphing lines and checking if points lie on lines are detailed
• Formulas for midpoint, segment length, and line equations are provided
• Techniques for finding slope (coefficiente angolare) are outlined
• Distance from a point to a line is briefly covered

19/09/2022

1021

 

2ªl

 

Matematica

55

ye
y
2 quad. + 1'quad (+;+)
(-;+)
3° quod.
(-;-)
piamo cartesiano
ex:
COME PASSARE DALLA FORMA IMPLICITA A QUELLA ESPLICITA:
+
ax+by+C =0
1.

Midpoint Formula and Segment Length

This page focuses on finding the midpoint of a line segment and calculating segment length in the coordinate plane.

The midpoint formula is introduced: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

For segment length, the distance formula is given: AB = √[(x₂ - x₁)² + (y₂ - y₁)²]

Several examples demonstrate the application of these formulas.

The page then transitions to equations of lines, presenting two main methods:

  1. Using two points on the line (point-point form)
  2. Using a point and the slope (point-slope form)

Vocabulary: The punto medio (midpoint) is the point that divides a line segment into two equal parts.

Example: For a segment with endpoints A(2,1) and B(-3,5), the midpoint M is calculated as (-1/2, 3).

Definition: The length of a line segment is the distance between its endpoints, calculated using the distance formula derived from the Pythagorean theorem.

Highlight: The midpoint and length formulas are fundamental tools in coordinate geometry, used in many advanced problems and proofs.

ye
y
2 quad. + 1'quad (+;+)
(-;+)
3° quod.
(-;-)
piamo cartesiano
ex:
COME PASSARE DALLA FORMA IMPLICITA A QUELLA ESPLICITA:
+
ax+by+C =0
1.

Slope and Distance to a Line

This page delves deeper into finding the slope (coefficiente angolare) of a line and introduces the concept of distance from a point to a line.

The slope formula is revisited: m = (y₂ - y₁) / (x₂ - x₁)

Examples are provided for calculating slope given two points.

The point-slope form of a line equation is emphasized: y - y₁ = m(x - x₁)

The page concludes with the formula for the distance from a point to a line: d = |ax₀ + by₀ + c| / √(a² + b²)

Where (x₀, y₀) is the point and ax + by + c = 0 is the equation of the line in implicit form.

Definition: The coefficiente angolare (slope) of a line measures its steepness and direction, represented by m in the equation y = mx + q.

Example: For a line passing through A(2,-3) and B(-1,4), the slope is calculated as m = (4 - (-3)) / (-1 - 2) = 7 / -3 = -7/3.

Highlight: Understanding slope is crucial for analyzing the relationship between lines, including parallel and perpendicular lines.

Vocabulary: The distance from a point to a line is the length of the perpendicular segment from the point to the line.

ye
y
2 quad. + 1'quad (+;+)
(-;+)
3° quod.
(-;-)
piamo cartesiano
ex:
COME PASSARE DALLA FORMA IMPLICITA A QUELLA ESPLICITA:
+
ax+by+C =0
1.

Converting Between Implicit and Explicit Forms

This page explains how to convert a linear equation from forma implicita to forma esplicita. The process involves isolating the y-variable on one side of the equation.

The general steps are:

  1. Start with the implicit form ax + by + c = 0
  2. Subtract ax and c from both sides
  3. Factor out b from the y term
  4. Divide both sides by b to isolate y

An example is given: 2x + 3y - 6 = 0 is converted to y = -2/3x + 2

The page also covers:

  • Graphing lines using a point-slope approach
  • Checking if a point lies on a line by substituting coordinates
  • Special cases of vertical and horizontal lines

Definition: The forma implicita of a line is ax + by + c = 0, while the forma esplicita is y = mx + q, where m is the slope and q is the y-intercept.

Example: To check if point P(-2,1) lies on y = -3x + 5, substitute the x-coordinate: 1 ?= -3(-2) + 5. Since 1 = 1, the point does lie on the line.

Highlight: Understanding how to convert between implicit and explicit forms is crucial for solving many geometry problems involving lines.

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.

Ranked #1 Education App

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

15 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.