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Parallel & Perpendicular Lines | Mathematics for GCSE/IGCSE - Year 11 PDF Download
Parallel Lines
What are parallel lines?
- Parallel lines share the same slope but are distinct from one another.
- Parallel lines never intersect.
- When lines are expressed in the form y = mx + c, you can easily identify parallel lines by their identical 𝑚m values (slopes).
- For example, y = 3x + 7 and y = 3x − 4 are parallel because they have the same slope.
- In contrast, y = 2x + 3 and y = 3x + 3 are not parallel since their slopes are different.
- Although y = 4x + 9 and y = 4x + 9 have the same slope, they are not parallel because they represent the same line.
How do I find the equation of a line parallel to another line?
- Since parallel lines have the same slope, a line in the form y = mx + c will be parallel to a line in the form y = mx + d, with the same 𝑚m for both lines.
- If c = d, then the lines would be identical and not parallel.
- To find the equation of a line parallel to y = mx + c, you will need information about a point (x1, y1) that the parallel line y = mx + d passes through.
- Substitute the point (x1, y1) into y = mx + d and solve for d.
Perpendicular Lines
- It is a known fact that parallel lines have equal gradients.
- Perpendicular lines intersect each other at right angles, specifically at 90°.
What’s the deal with perpendicular gradients (and lines)?
- Before delving into perpendicular gradients and lines, it's crucial to grasp how to determine the equation of a straight line. This skill will be fundamental for solving related problems.
- Gradients m1 and m2 are considered perpendicular if m1 x m2 = -1.
- For instance,
- 1 and −1
- 1/3 and 3
- We can use m2 = −1 ÷ m1 to find a perpendicular gradient. This is called the negative reciprocal.
- When in doubt, sketching the lines can provide a visual aid in understanding the perpendicular relationship between two gradients.
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FAQs on Parallel & Perpendicular Lines - Mathematics for GCSE/IGCSE - Year 11
| 1. How can we determine if two lines are parallel using their equations? |  |
Ans. Two lines are parallel if their slopes are equal. This means that if the gradients of the lines are the same, then the lines are parallel.
| 2. Can two perpendicular lines have the same slope? | |
Ans. No, perpendicular lines have slopes that are negative reciprocals of each other. This means that if one line has a slope of m, the perpendicular line will have a slope of -1/m.
| 3. How can we find the equation of a line that is perpendicular to a given line passing through a specific point? | |
Ans. To find the equation of a line perpendicular to a given line, first find the negative reciprocal of the slope of the given line. Then, use the point-slope form of the equation to find the equation of the perpendicular line passing through the given point.
| 4. Can two lines be both parallel and perpendicular to each other at the same time? | |
Ans. No, two lines cannot be both parallel and perpendicular to each other at the same time. If two lines are parallel, they will never intersect, while perpendicular lines intersect at a right angle.
| 5. How can we determine if two lines are perpendicular by looking at their equations? | |
Ans. Two lines are perpendicular if the product of their slopes is -1. This means that if the gradients of the lines are m1 and m2, then m1 * m2 = -1, indicating that the lines are perpendicular.
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Dec 22, 2024 Last updated |
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