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Perpendicular lines are straight lines that bisect each other at an angle of 90°. When a straight line makes an exact angle with another line then it will be called a perpendicular line. Square, rectangle & right triangle have perpendicular sides. In the perpendicular line, the formula is used to find out the slope of the two lines. A straight line that passes through another one at an exact angle is said to be perpendicular to another one. These lines have ropes that are contrary to reciprocals of each other.
Key Terms: Perpendicular Lines, Slope, Straight Lines, Triangles, Coordinates
What is a Perpendicular Line?
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A perpendicular line is consecutive and makes a perfect angle along with the other line. It is marked as a little square between these title lines. These lines are intersecting each other and the angle between these lines is 90°. The angle is always a right angle.
Perpendicular Lines
When two lines intersect each other at an angle of 90°, they are said to be perpendicular to each other. The perpendicular line is represented with the symbol (⊥).
Some examples of perpendicular lines in daily life are floor tiles, fences, traffic signs & furniture, corners of two walls, Red cross symbol, etc.
Examples of Perpendicular Line
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Properties of a Perpendicular Line
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The properties of a perpendicular line are as follows:
- The perpendicular lines always meet each other at an obvious angle.
- The angle between the perpendicular lines is always a straight angle of 90°.
- Two perpendicular lines are parallel to each other if they are perpendicular to the same line.
- The contiguous sides of a rectangle or square are always perpendicular to each other.
- The sides of a right-angled triangle always enclose the right angle. Likewise, they are perpendicular to each other.
- If the slopes of two lines are negative or conversely reciprocals of each other, they are said to be perpendicular.
Perpendicular Lines
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Formula of Perpendicular Line
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Perpendicular line makes an angle with a particular point by which the line passes. The equation & coordinates are prerequisites for identifying the perpendicular line.
Let’s consider the equation as
ax + by + c = 0
The coordinates are;
(x1, y1)
The slope should be a/b.
So, if one line is perpendicular to this line then the product of the slope should be -1.
If m1 & m2 are the slopes of the two lines & are perpendicular to each other, then their product will be -1.
Perpendicular lines = m1 × m2 = -1
The slope of the line m = -a/b
The perpendicular line equation:
(y - y1) = m (x - x1)
Slope of a Perpendicular Line
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Slope of perpendicular lines is the negative reciprocal of another line. It can be calculated with the given line. The product of the slope of a given line and slope of the perpendicular line is always equal to -1.
For example: If two lines AB & CD then perpendicular to each other then the slope of AB = m1 & the slope of CD = m2
So, it is clear that two lines are perpendicular to each other if the product of the slope of two lines is equal to minus of unity.
The formula for the slope:
m1.m2 = - 1
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Things to Remember
- The perpendicular lines perpetually cross each other at an angle of 90°.
- The two lines are perpendicular only if the slope of one line is the negative reciprocal of the other one.
- The perpendicular lines are represented by the symbol (⊥).
- The perpendicular line equation is given as; (y - y1) = m (x - x1).
- The perpendicular lines have a conversely reciprocal slope.
Sample Questions
Ques: Check out whether 3x +2y+5=0 & 2x – 3y + 1= 0 are perpendiculars or not? (3 marks)
Ans: As given in the question,
3x +2y +5 =0, 2x – 3y +1= 0
First, we need to find out the slopes of these lines.
Suppose if the product of their slope is -1 then they are perpendicular to each other.
The formula of slope: m1.m2 = -1
The slope of the first line = -3/2
The slope of the second line = -2 / -3 = 2/3
Product of the slopes = -3/2 2/3 = -1.
Thus, the lines are perpendicular as the product of their slopes is -1.
Ques: What will be the slope of a line that is perpendicular to the line 2x + 7y+ 5= 0. (5 marks)
Ans: The equation given in the question is;
2x+6y+5=0
The slope of the line is;
-2/6 = -1/3
Suppose that the slope of the perpendicular line is = m
From line 1 we can write,
m×(-1/3)= -1
m=3
Consequently, the slope of the perpendicular line is 3.
Ques: Write the rules of perpendicular lines. (3 marks)
Ans: These are some rules of perpendicular lines;
- Two lines are perpendicular if they are in the equivalent plane.
- Two lines are perpendicular if they cross each other at an angle of 90°.
- If two lines have the same slope then they are perpendicular.
- The angle between the lines should be the exact angle.
Ques: Find the equation of a line that is parallel to y = 3x +1 & the line is passing through the points (4,5). (5 marks)
Ans: The slope of the line, y = 3x + 1
Thus, m = 3
The required line also has a slope that is equal to 3.
Point = 4,5
Let’s substitute the values
y-y1= m (x-x1)
- y-4=3(x-5)
- y-4=3(x-5)
- y-4=3x-15
- y=3x-11
Hence, the equation for the parallel line y= 3x+1 will be y=3x-11
Ques: Find out the slope of a line perpendicular to the given line y = -3x + 8. (3 marks)
Ans: As per the equation,
The given equation is y = -3x + 8
The slope of the equation = -3
The slope of the perpendicular line m = the negative reciprocal of the given slope
m = -1/ (-3) = ⅓
Hence, the slope of the perpendicular line is m = 1/3.
Ques: Are all the intersecting lines perpendicular? (2 marks)
Ans: No, it is not true because not all the intersecting lines are perpendicular. They might not be perpendicular. Two lines are perpendicular only if they intersect with each other at a right angle. But if two lines intersect with each other at different angles then they are not perpendicular.
Ques: If a line passes through the points (0,3) & (-1,-5) & another line passes through the points (2,0) & (-2,1). Find out if these lines are perpendicular or not? (3 marks)
Ans: The slope of first line;
m1=(-5+3)/-1=-2/-1=2
m2= (1-0)/(2-2)=1/(-4)= -1/2
Hence, m1.m2= 2 ×(-1/2)=-1
These lines are perpendicular to the given line.
Ques: check out whether the lines are perpendicular or not? If these equations 5y + 1 = 5x & x + 3y are given. (3 marks)
Ans: Let’s write both of these equations in the slope interlope-intercept form:
5y+1=5x
- 5y=5x-1
- Y=x-1/5
- Slope x+3=y
- y=x+3
- Slope,=1
Hence, both of the slopes are the same. So, these lines are not perpendicular.
Ques: What is the product of the slope of a perpendicular line? (2 marks)
Ans: The product of the slope of a perpendicular line towards the slope of the original line is always equal to -1.
Slope of a perpendicular line = m1.m2 = -1
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