IB MYP 4-5 Maths- Parallel and perpendicular lines - Study Notes - New Syllabus
IB MYP 4-5 Maths- Parallel and perpendicular lines – Study Notes
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- Parallel and perpendicular lines
IB MYP 4-5 Maths- Parallel and perpendicular lines – Study Notes – All topics
Parallel and perpendicular lines
Parallel Lines in Linear Functions
Two lines are parallel if they never meet and have the same slope. This means they go in the same direction forever.
Condition for Parallel Lines:
Same value of \( m \) in their equations \( y = mx + c \)
- The only difference is in their y-intercepts (the value of \( c \)).
- They are equidistant at all points.
Note:
- If two lines have equations \( y = mx + c_1 \) and \( y = mx + c_2 \), then they are parallel if \( c_1 \neq c_2 \).
- Parallel lines do not intersect (unless they are the same line).
Example :
Are the lines \( y = 3x + 4 \) and \( y = 3x – 7 \) parallel?
▶️ Answer/Explanation
Both have the same coefficient of \( x \) (3).
Result: Yes, they are parallel because their slopes are the same.
Example :
Find the equation of the line parallel to \( y = 2x + 5 \) and passing through (1, 4).
▶️ Answer/Explanation
Since the new line is parallel, it has the same coefficient of \( x \): \( y = 2x + c \).
Substitute (1, 4): \( 4 = 2(1) + c \Rightarrow 4 = 2 + c \Rightarrow c = 2 \).
Final Answer: \( y = 2x + 2 \).
Perpendicular Lines in Linear Functions
Two lines are perpendicular if they intersect at a right angle (90°).
Condition for Perpendicular Lines:
- If one line has equation \( y = m_1x + c_1 \), the other has \( y = m_2x + c_2 \) where \( m_2 = -\dfrac{1}{m_1} \).
- They intersect at a right angle (90°).
Note:
- Perpendicularity is about orientation: one line “cuts” the other at 90°.
- They often appear in coordinate geometry, design, and construction problems.
Example :
Are the lines \( y = 2x + 3 \) and \( y = -\dfrac{1}{2}x + 5 \) perpendicular?
▶️ Answer/Explanation
Check: \( (2) \times \left(-\dfrac{1}{2}\right) = -1 \).
Result: Yes, the lines are perpendicular.
Example :
Find the equation of the line perpendicular to \( y = 3x + 2 \) and passing through (2, -1).
▶️ Answer/Explanation
For the original line, the coefficient of \( x \) is 3, so the perpendicular line has a coefficient of \( -\dfrac{1}{3} \).
Equation: \( y = -\dfrac{1}{3}x + c \).
Substitute (2,-1): \( -1 = -\dfrac{2}{3} + c \Rightarrow c = -\dfrac{1}{3} \).
Final Answer: \( y = -\dfrac{1}{3}x – \dfrac{1}{3} \).