IB MYP 4-5 Maths- Gradients of perpendicular lines- Study Notes - New Syllabus
IB MYP 4-5 Maths- Gradients of perpendicular lines – Study Notes
Extended
- Gradients of perpendicular lines
IB MYP 4-5 Maths- Gradients of perpendicular lines – Study Notes – All topics
Perpendicular Lines
Perpendicular Lines
Two lines are perpendicular if they intersect at a right angle (90°). The product of their gradients is:
\( m_1 \times m_2 = -1 \)
This means if one line has a gradient \( m_1 \), then the perpendicular line has a gradient:
\( m_2 = -\dfrac{1}{m_1} \)
Key Points:
- Condition: \( m_1 \cdot m_2 = -1 \).
- If the first line is horizontal (\( m = 0 \)), the perpendicular line is vertical (undefined slope).
- Perpendicular lines form a right angle at their intersection point.
Question:
Find the equation of a line that is perpendicular to \( y = 2x + 3 \) and passes through the point (4, 1).
▶️ Answer/Explanation
Step 1: Gradient of given line: \( m_1 = 2 \).
Step 2: Gradient of perpendicular line: \( m_2 = -\dfrac{1}{m_1} = -\dfrac{1}{2} \).
Step 3: Equation using point-slope form:
\( y – 1 = -\dfrac{1}{2}(x – 4) \)
\( y – 1 = -\dfrac{1}{2}x + 2 \)
\( y = -\dfrac{1}{2}x + 3 \)
Question:
A line passes through (0, 5) and is perpendicular to the line joining (2, 3) and (6, 7). Find its equation.
▶️ Answer/Explanation
Step 1: Find gradient of line through (2, 3) and (6, 7):
\( m_1 = \dfrac{7 – 3}{6 – 2} = \dfrac{4}{4} = 1 \).
Step 2: Gradient of required line:
\( m_2 = -\dfrac{1}{m_1} = -1 \).
Step 3: Use point (0, 5):
\( y – 5 = -1(x – 0) \)
\( y = -x + 5 \)
Question :
Find the equation of the perpendicular bisector of the segment joining A(2, -1) and B(6, 3).
▶️ Answer/Explanation
Step 1: Find midpoint of AB:
\( M = \left( \dfrac{2+6}{2}, \dfrac{-1+3}{2} \right) = (4, 1) \).
Step 2: Gradient of AB:
\( m_1 = \dfrac{3 – (-1)}{6 – 2} = \dfrac{4}{4} = 1 \).
Step 3: Gradient of perpendicular bisector:
\( m_2 = -\dfrac{1}{m_1} = -1 \).
Step 4: Equation using point (4, 1):
\( y – 1 = -1(x – 4) \)
\( y – 1 = -x + 4 \)
\( y = -x + 5 \)