Parallel and Perpendicular Lines
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Parallel and Perpendicular Lines — Free MYP5 Mathematics (Standard) Practice Questions

1QuestionGradient of Parallel LinesConcept Practice
2 marks~3 minCriterion A
A city planner is designing two parallel roads on a grid map. Road A passes through the points (2,3)(2, 3) and (6,11)(6, 11).
a
Calculate the gradient of Road A. [1]
b
Road B is parallel to Road A and passes through the point (0,5)(0, 5). The planner claims Road B will intersect the yy-axis at a height of 5 map units, representing 50 000 cm in the real world (scale 1:10 0001 : 10\,000). Justify whether this claim is mathematically consistent. [1]
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2QuestionGradient of Parallel LinesAssessment Practice
6 marks~9 minCriterion D
A civil engineer checks whether a proposed road segment is parallel to an existing road modelled by y=34x−2y = \dfrac{3}{4}x - 2, where xx and yy are horizontal and vertical distances in kilometres.

Survey data for the proposed segment gives two control points:

First control point: (1, 1.8)(1,\ 1.8)
Last control point: (7, 6.0)(7,\ 6.0)
a
State the gradient formula and calculate the gradient of the proposed road segment using the two control points. [3]
b
Deduce whether the proposed road segment is parallel to the existing road, justifying your answer by comparing the two gradients. [1]
c
The engineer requires the two roads to be parallel to within a tolerance of ±0.02\pm 0.02. Advise the engineer whether the proposed segment meets this requirement, and suggest one adjustment to the last control point to achieve exact parallelism. [2]
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Solutions

3QuestionGradient of Perpendicular LinesConcept Practice
2 marks~3 minCriterion C
An urban planner is designing a city block. Two roads must be built so that they intersect at exactly 90°.

Road A follows the line y=3x−5y = 3x - 5.

Road B passes through the point (6,1)(6, 1) and must be perpendicular to Road A.
a
State the gradient of Road B. [1]
b
The planner claims Road B can be represented by y=−13x+3y = -\dfrac{1}{3}x + 3. Justify whether this equation is correct and whether the road design meets the 90° requirement. [1]

Solutions

4QuestionVerifying Line Relationships AlgebraicallyAssessment Practice
6 marks~9 minCriterion B
An urban planner is designing a city block. Two roads are being built: Road A has slope m1=34m_1 = \dfrac{3}{4} and Road B has slope m2=−43m_2 = -\dfrac{4}{3}. A third road, Road C, runs parallel to Road A with slope m3=34m_3 = \dfrac{3}{4}.
a
Calculate m1×m2m_1 \times m_2 and m1×m3m_1 \times m_3. [2]
b
Deduce the general algebraic relationship between the slopes of two perpendicular lines and between the slopes of two parallel lines. [2]
c
The planner claims Roads A and B form a perfect right-angle intersection, required by the city's safety code. Justify whether this claim is correct, and advise the planner on what adjustment to Road B's slope would be needed if it were instead built with slope m=−54m = -\dfrac{5}{4}. [2]

Solutions

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