Introduction to perpendicular line segments:
Perpendicular line:
In geometry, two line segments or planes (or a line and a plane), are considered perpendicular (or orthogonal) to each other if they form congruent adjacent angles (a T-shape). Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B.
Fig(i) Perpendicular line
Line segments:
The line segments are the the part of the line that has two end points.It include all the points between its end points.For example,consider the following figure,
Fig(ii) Line segment
In the above figure AB is a line and CD is a line segment.C and D are the two end points of a line segment.We are going to see about the perpendicular line and line segments.
Examples for Perpendicular Line Segments:
The following are the geometry examples for line segments,
Sides of the triangle
Sides of the Rectangle
Sides of the Square
Properties of perpendicular line:
The multiplication of slopes of the perpendicular lines is equal to -1.
That is m1=slope of perpendicular line 1
m2 = Slope of perpendicular line 2
m1 × m2 = -1
Distance formula for line segments:
The following formula used to find the lengh of the line segments ,
length = `sqrt((x2-x1)^2+(y2-y1)^2))`
Here (x1,y1) and (x2,y2) are the two end points of the line segments.
Problems on Perpendicular Line Segments:
Problem 1:
Find the equation of a line which is perpendicular to 4y - x = 20 and passes through the point (2, -3).
Solution:
Given 4y - x = 20 and the point (2,-3)
To find the perpendicular line we need to find the slope.
To find the slope we need to change the given equation into slope intercept form.
4y - x = 20
Add x on both side,
4y - x = 20
+ x = +x
4y = x + 20
Divide by 4 on both side,
y = (x/4) + 5
y = mx + b form
So the slope m = 1/4
We know that multiple of slopes of the perpendicular lines equals to -1
1/4 * m = -1
Multiply by 4 on both sides,
m = -4
The line equation is,
(y-y1) = m(x-x1)
(y - (-3)) = -4(x-2)
y+4 = -4x+8
Subtract 4 on both sides,
y = -4x +4
Answer : The line perpendicular to the given line is y = -4x +4
Problem 2:
By using the line segments distance formula find the length of the line segmen xy with x(5 ,3) y(1,2)
Solution:
length = `sqrt((x2-x1)^2+(y2-y1)^2))`
x1 = 5 y1 =3 x2 = 1 y2 = 2
=` sqrt( ( 1-5)^2 + ( 2-3)^2)`
=` sqrt( (-4)^2 + (-1)^2)`
= `sqrt (16+1)`
= `sqrt (17)`
Answer :The length of the line segment xy = `sqrt (17)`
Perpendicular line:
In geometry, two line segments or planes (or a line and a plane), are considered perpendicular (or orthogonal) to each other if they form congruent adjacent angles (a T-shape). Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B.
Fig(i) Perpendicular line
Line segments:
The line segments are the the part of the line that has two end points.It include all the points between its end points.For example,consider the following figure,
Fig(ii) Line segment
In the above figure AB is a line and CD is a line segment.C and D are the two end points of a line segment.We are going to see about the perpendicular line and line segments.
Examples for Perpendicular Line Segments:
The following are the geometry examples for line segments,
Sides of the triangle
Sides of the Rectangle
Sides of the Square
Properties of perpendicular line:
The multiplication of slopes of the perpendicular lines is equal to -1.
That is m1=slope of perpendicular line 1
m2 = Slope of perpendicular line 2
m1 × m2 = -1
Distance formula for line segments:
The following formula used to find the lengh of the line segments ,
length = `sqrt((x2-x1)^2+(y2-y1)^2))`
Here (x1,y1) and (x2,y2) are the two end points of the line segments.
Problems on Perpendicular Line Segments:
Problem 1:
Find the equation of a line which is perpendicular to 4y - x = 20 and passes through the point (2, -3).
Solution:
Given 4y - x = 20 and the point (2,-3)
To find the perpendicular line we need to find the slope.
To find the slope we need to change the given equation into slope intercept form.
4y - x = 20
Add x on both side,
4y - x = 20
+ x = +x
4y = x + 20
Divide by 4 on both side,
y = (x/4) + 5
y = mx + b form
So the slope m = 1/4
We know that multiple of slopes of the perpendicular lines equals to -1
1/4 * m = -1
Multiply by 4 on both sides,
m = -4
The line equation is,
(y-y1) = m(x-x1)
(y - (-3)) = -4(x-2)
y+4 = -4x+8
Subtract 4 on both sides,
y = -4x +4
Answer : The line perpendicular to the given line is y = -4x +4
Problem 2:
By using the line segments distance formula find the length of the line segmen xy with x(5 ,3) y(1,2)
Solution:
length = `sqrt((x2-x1)^2+(y2-y1)^2))`
x1 = 5 y1 =3 x2 = 1 y2 = 2
=` sqrt( ( 1-5)^2 + ( 2-3)^2)`
=` sqrt( (-4)^2 + (-1)^2)`
= `sqrt (16+1)`
= `sqrt (17)`
Answer :The length of the line segment xy = `sqrt (17)`
