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 tusharparik • 14 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
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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