Construction and Loci

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Construction and Loci — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionConstructing Perpendicular and Angle BisectorsConcept Practice

2 marks~3 minCriterion B

The diagrams below show a series of angle bisector constructions. The first diagram shows an angle of 60 degrees with one bisector dividing it into two 30-degree angles. The second diagram shows the same angle with two bisectors, dividing it into four 15-degree angles. The third diagram shows the angle with four bisectors, dividing it into eight 7.5-degree angles.

Outline the visual pattern in the number of bisectors as the construction continues. Predict how many bisectors the next diagram in the sequence would have.

Solutions

2QuestionUsing Compass and Ruler for Geometric ConstructionsAssessment Practice

2 marks~3 minCriterion C

A city planner must build a straight path through a rectangular park, connecting the midpoints of the two shorter sides so that the path is equidistant from both longer sides.

Figure 1 shows rectangle

ABCD

, where

AB

and

CD

are the shorter sides.

Construct the path using only a compass and ruler. Describe each step of your construction. [1]

The park is later found to be a trapezoid, not a rectangle. Advise the city planner whether this construction still meets the design requirement, justifying your answer with a geometric reason. [1]

Solutions

3QuestionConstructing Triangles with Given ConditionsConcept Practice

4 marks~6 minCriterion A

A surveyor is planning a triangular plot of land. Two boundary lines meet at point

A

: one runs

5\ \text{cm}

(on the scale drawing) to point

B

, and the other runs at

60°

to point

C

, which is

7\ \text{cm}

from

A

Construct triangle

ABC

with

AB = 5\ \text{cm}

\angle BAC = 60°

, and

AC = 7\ \text{cm}

. Label all three vertices. [1]

Measure

BC

to the nearest millimetre and record your answer in centimetres. [1]

The surveyor states that the plot is valid only if all three boundary lengths are less than

6.5\ \text{cm}

. Advise the surveyor whether this plot meets the requirement, referring to each boundary length in your response. [2]

Solutions

4QuestionConstructing Perpendicular and Angle BisectorsAssessment Practice

6 marks~9 minCriterion D

A self-driving boat navigates a narrow channel with an irregularly shaped island. Its sensors continuously identify the two nearest points on opposite sides of the channel — one on the island's perimeter, one on the channel wall — and the boat must remain equidistant from both to avoid collision. The channel is 50 m wide at its narrowest point.

Explain how perpendicular bisectors can model the boat's safe navigation path through the channel. [2]

Discuss two limitations of applying this perpendicular bisector model in a real-world maritime environment. [2]

The boat's sensors have a positional error of

\pm 1.5

m. Advise the boat's navigation engineers whether the perpendicular bisector path alone provides a sufficient safety margin at the 50 m narrowest point, and justify what additional measures, if any, are required for safe passage. [2]

Solutions

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