Can you help me to summarize all possible equation and formulas with example for Chapter 13 Section and Mid-POint formula & Chapter 14 Equation of Line. I am having trouble in understanding it make it easy.

Asked by tusharparik14 days ago

Can you help me to summarize all possible equation and formulas with example for Chapter 13 Section and Mid-POint formula & Chapter 14 Equation of Line. I am having trouble in understanding it make it easy.

Answers

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14 days ago
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Mathematics - Chapter 13 & 14 Formulas and Equations

Here's a summary of the equations and formulas you'll need for Chapter 13 (Section & Midpoint Formulas) and Chapter 14 (Equation of a Line). I'll break it down to make it easier to understand.

Chapter 13: Section and Midpoint Formulas

Section Formula

The section formula helps you find the coordinates of a point that divides a line segment in a given ratio.

Section Formula (Internal Division)

If point P(x, y) divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n, then:
x = (mx₂ + nx₁) / (m + n)
y = (my₂ + ny₁) / (m + n)

Section Formula (External Division)

If point P(x, y) divides the line segment joining A(x₁, y₁) and B(x₂, y₂) externally in the ratio m:n, then:
x = (mx₂ - nx₁) / (m - n)
y = (my₂ - ny₁) / (m - n)

Example (Internal Division)

Find the coordinates of the point that divides the line segment joining A(1, 2) and B(4, 5) internally in the ratio 2:1.
Here, x₁ = 1, y₁ = 2, x₂ = 4, y₂ = 5, m = 2, n = 1.
x = (2*4 + 1*1) / (2 + 1) = (8 + 1) / 3 = 3
y = (2*5 + 1*2) / (2 + 1) = (10 + 2) / 3 = 4
So, the coordinates are (3, 4).

Midpoint Formula

The midpoint formula is a special case of the section formula where the point divides the line segment in the ratio 1:1. It finds the exact middle point of a line segment.

Midpoint Formula

If M(x, y) is the midpoint of the line segment joining A(x₁, y₁) and B(x₂, y₂), then:
x = (x₁ + x₂) / 2
y = (y₁ + y₂) / 2

Example (Midpoint)

Find the midpoint of the line segment joining A(2, 3) and B(6, 7).
Here, x₁ = 2, y₁ = 3, x₂ = 6, y₂ = 7.
x = (2 + 6) / 2 = 8 / 2 = 4
y = (3 + 7) / 2 = 10 / 2 = 5
So, the midpoint is (4, 5).

Chapter 14: Equation of a Line

Equation of a Line

The equation of a line represents the relationship between the x and y coordinates of all the points on that line. There are several forms of the equation.

Slope-Intercept Form

This form is useful when you know the slope (m) and the y-intercept (c) of the line.
y = mx + c
Where 'm' is the slope and 'c' is the y-intercept.

Point-Slope Form

This form is useful when you know the slope (m) and a point (x₁, y₁) on the line.
y - y₁ = m(x - x₁)

Two-Point Form

This form is useful when you know two points (x₁, y₁) and (x₂, y₂) on the line.
First, find the slope: m = (y₂ - y₁) / (x₂ - x₁)
Then, use the point-slope form with either point.

General Form

The general form of a linear equation is:
Ax + By + C = 0
Where A, B, and C are constants.

Example (Slope-Intercept Form)

Find the equation of a line with a slope of 2 and a y-intercept of -1.
Using y = mx + c:
y = 2x - 1

Example (Point-Slope Form)

Find the equation of a line with a slope of 3 that passes through the point (1, 2).
Using y - y₁ = m(x - x₁):
y - 2 = 3(x - 1)
y - 2 = 3x - 3
y = 3x - 1

Study Tip: Practice converting between these forms to become comfortable with linear equations. Remember that the slope (m) represents the steepness of the line, and the y-intercept (c) is where the line crosses the y-axis.

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