Parallel and perpendicular lines

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Parallel lines

Distinct non-vertical lines with equal gradients and different intercepts are parallel. Vertical lines are parallel to one another but have undefined gradient.
y = 3x + 1 and y = 3x - 4 are parallel (both gradient 3).
To find the line parallel to y = 2x + 5 through (1, 9): use m = 2, then 9 = 2(1) + c gives c = 7, so y = 2x + 7.

Same gradient, different intercepts

Perpendicular lines

For two non-vertical lines meeting at a right angle, m₁m₂ = −1. Horizontal and vertical lines are also perpendicular, but the product rule cannot use an undefined gradient.
The perpendicular gradient is the negative reciprocal: flip the fraction and change the sign.
A line with gradient 2 has a perpendicular gradient -\frac{1}{2}; gradient -\frac{3}{4} gives \frac{4}{3}.

Negative reciprocal gradients

Rearranging first

Always rearrange to y = mx + c before reading the gradient.
2y = 6x + 4 becomes y = 3x + 2, so m = 3. x + 2y = 8 becomes y = -\frac{1}{2}x + 4, so m = -\frac{1}{2}.

Finding the equation of a parallel or perpendicular line

Step 1: find the required gradient. Step 2: substitute the given point into y = mx + c to find c.
The line perpendicular to y = 2x + 1 through (4, 3): m = -\frac{1}{2}; 3 = -\frac{1}{2}(4) + c gives c = 5, so y = -\frac{1}{2}x + 5.

Special cases

Horizontal lines (y = k) are perpendicular to vertical lines (x = k); their gradients are 0 and undefined.
Do not take the reciprocal of zero: a line perpendicular to a horizontal line is vertical, with equation x = k.

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Câu hỏi luyện tập

Xem trước miễn phí — 8 trên 32 câu hỏi. Đăng ký để xem tất cả.

1.What is true of the gradients of distinct non-vertical parallel lines?
Easy
- AThey add to zero
- BThey are equal
- CThey multiply to -1
- DThey differ by 1
2.Which line is parallel to y = 5x + 3?
Easy
- Ay = 3x + 5
- By = 5x - 7
- Cy = -5x + 3
- Dy = \frac{x}{5}
3.What is the gradient of a line parallel to y = -3x + 1?
Easy
- A3
- B-3
- C\frac{1}{3}
- D-\frac{1}{3}
4.What is the gradient of a line perpendicular to y = 4x?
Easy
- A4
- B-4
- C\frac{1}{4}
- D-\frac{1}{4}
5.What is the gradient of a line perpendicular to y = -\frac{2}{5}x?
Easy
- A\frac{2}{5}
- B-\frac{5}{2}
- C\frac{5}{2}
- D-\frac{2}{5}
6.Which pair of lines is perpendicular?
Easy
- Ay = 3x and y = 3x + 1
- By = 3x and y = -\frac{1}{3}x
- Cy = 3x and y = -3x
- Dy = 3x and y = \frac{1}{3}x
7.What is the gradient of y = 6 - 2x?
Easy
- A6
- B-2
- C2
- D-6
8.What is the gradient of 2y = 8x - 6?
Medium
- A8
- B4
- C2
- D-3

Thêm câu hỏi

9.What is the gradient of x + 2y = 10?
Medium
- A1
- B\frac{1}{2}
- C-\frac{1}{2}
- D-2
10.Find the equation of the line parallel to y = 3x + 1 through (2, 10).
Medium
- Ay = 3x + 4
- By = 3x + 10
- Cy = 3x - 4
- Dy = x + 8
11.Find the equation of the line parallel to y = -2x through (1, 5).
Medium
- Ay = -2x + 7
- By = -2x + 3
- Cy = 2x + 3
- Dy = -2x + 5
12.Find the equation of the line perpendicular to y = 2x + 1 through (4, 3).
Medium
- Ay = -\frac{1}{2}x + 5
- By = -2x + 11
- Cy = \frac{1}{2}x + 1
- Dy = -\frac{1}{2}x + 3
13.Line A has equation y = 3x - 2. Line B is perpendicular to A and passes through (6, 1). What is B's equation?
Medium
- Ay = -\frac{1}{3}x + 3
- By = -3x + 19
- Cy = \frac{1}{3}x - 1
- Dy = -\frac{1}{3}x + 1
14.Find the equation of the perpendicular bisector of A(0, 2) and B(4, 4).
Medium
- Ay = -2x + 7
- By = -2x + 6
- Cy = \frac{1}{2}x + 2
- Dy = 2x - 3
15.Which of these lines is parallel to the x-axis?
Easy
- Ax = 5
- By = 5
- Cy = x
- Dy = 5x
16.Which line is perpendicular to y = 4?
Easy
- Ay = -4
- Bx = 2
- Cy = x
- Dy = 4x
17.Lines y = kx + 3 and y = 6x - 1 are parallel. What is k?
Medium
- A3
- B6
- C-6
- D\frac{1}{6}
18.Lines y = kx and y = 4x + 1 are perpendicular. What is k?
Medium
- A4
- B-4
- C\frac{1}{4}
- D-\frac{1}{4}
19.Rearrange 3y = 12x + 9 into y = mx + c.
Easy
- Ay = 4x + 3
- By = 12x + 3
- Cy = 4x + 9
- Dy = 36x + 27
20.Which line is parallel to 2y = 4x + 5?
Medium
- Ay = 4x
- By = 2x + 7
- Cy = \frac{1}{2}x
- Dy = -2x
21.Are y = 2x + 1 and 4x - 2y = 7 parallel?
Medium
- AYes, both have gradient 2
- BNo, gradients differ
- CYes, both have gradient 4
- DNo, they are perpendicular
22.What is the gradient of the line through (0, 0) and (3, 6), and of a line perpendicular to it?
Medium
- A2 and -\frac{1}{2}
- B2 and 2
- C3 and -\frac{1}{3}
- D\frac{1}{2} and -2
23.Distinct non-vertical parallel lines have equal gradients.
Easy
True or false?
24.For two non-vertical perpendicular lines, the product of their gradients is −1.
Easy
True or false?
25.A line with gradient 3 is perpendicular to a line with gradient -3.
Easy
True or false?
26.y = 2x + 1 and y = 2x - 5 are parallel.
Easy
True or false?
27.y = x and y = -x are perpendicular.
Medium
True or false?
28.Which lines are parallel to y = 3x - 2?
Medium
- Ay = 3x
- By = 3x + 8
- Cy = -3x
- Dy = \frac{x}{3}
- Ey = x + 3
29.Which lines are perpendicular to y = 2x?
Medium
- Ay = -\frac{1}{2}x
- By = -\frac{1}{2}x + 7
- Cy = 2x + 1
- Dy = -2x
- Ey = \frac{1}{2}x
30.Which line is parallel to y = x + 7?
Easy
- Ay = 7x
- By = x - 3
- Cy = -x + 7
- Dy = \frac{x}{7}
31.What is the negative reciprocal of 5?
Easy
- A5
- B-5
- C\frac{1}{5}
- D-\frac{1}{5}
32.Find the equation of the line through (2, 1) perpendicular to y = 4x - 3.
Medium

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