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Coordinate Geometry - Parallel and Perpendicular Lines

Grade 10IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The gradient (m) measures the steepness of a line.

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Parallel lines have the same gradient because they have the same steepness (m1 = m2).

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Perpendicular lines meet at a 90-degree angle; their gradients are negative reciprocals of each other (m1 * m2 = -1).

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The y-intercept (c) is the point where the line crosses the y-axis (x = 0).

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To find the equation of a line, you need at least one point on the line and the gradient.

📐Formulae

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} (Gradient formula)

y=mx+cy = mx + c (Gradient-intercept form)

m1=m2m_1 = m_2 (Condition for Parallel Lines)

m1×m2=−1m_1 \times m_2 = -1 or m2=−1m1m_2 = -\frac{1}{m_1} (Condition for Perpendicular Lines)

y−y1=m(x−x1)y - y_1 = m(x - x_1) (Point-gradient form)

💡Examples

Problem 1:

Find the equation of the line parallel to y=3x−5y = 3x - 5 that passes through the point (2,10)(2, 10).

Solution:

y=3x+4y = 3x + 4

Explanation:

  1. Since the lines are parallel, they have the same gradient, so m=3m = 3. 2. Use the point (2,10)(2, 10) in the equation y=mx+cy = mx + c: 10=3(2)+c10 = 3(2) + c. 3. Solve for cc: 10=6+c⇒c=410 = 6 + c \Rightarrow c = 4. 4. Write the final equation: y=3x+4y = 3x + 4.

Problem 2:

Line L1 has the equation y=−2x+7y = -2x + 7. Find the equation of line L2 which is perpendicular to L1 and passes through the point (−4,1)(-4, 1).

Solution:

y=12x+3y = \frac{1}{2}x + 3

Explanation:

  1. The gradient of L1 is m1=−2m_1 = -2. 2. The perpendicular gradient m2m_2 is the negative reciprocal: m2=−1/(−2)=1/2m_2 = -1 / (-2) = 1/2. 3. Use y−y1=m(x−x1)y - y_1 = m(x - x_1) with point (−4,1)(-4, 1): y−1=12(x−(−4))y - 1 = \frac{1}{2}(x - (-4)). 4. Simplify: y−1=12x+2⇒y=12x+3y - 1 = \frac{1}{2}x + 2 \Rightarrow y = \frac{1}{2}x + 3.

Problem 3:

Determine if the lines 2x+y=52x + y = 5 and 4x+2y=104x + 2y = 10 are parallel, perpendicular, or the same line.

Solution:

The lines are the same (coincident).

Explanation:

  1. Rearrange both into y=mx+cy = mx + c form. 2. Line 1: y=−2x+5y = -2x + 5. 3. Line 2: 2y=−4x+10⇒y=−2x+52y = -4x + 10 \Rightarrow y = -2x + 5. 4. Since both the gradient (m=−2m = -2) and the y-intercept (c=5c = 5) are identical, they represent the same line.