Construction and Loci

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

1QuestionConstructing Perpendicular and Angle BisectorsConcept Practice

2 marks~3 minCriterion A

A landscape architect is designing a triangular garden plot

ABC

. To install a straight irrigation pipe from vertex

B

to a point

D

on side

AC

, the pipe must bisect

\angle ABC

exactly, so that each flower bed on either side receives equal angular coverage from the sprinkler at

B

.

Deduce the relationship between

\angle ABD

and

\angle CBD

, and state the value of each angle if

\angle ABC = 74°

. [2]

Solutions

2QuestionConstructing Perpendicular and Angle BisectorsAssessment Practice

4 marks~6 minCriterion D

A farmer needs to divide an irregularly shaped, hilly field equally between two children. She marks two boundary points,

A

and

B

, on opposite edges of the field, then constructs the perpendicular bisector of segment

AB

as the proposed fence line.

Explain why the perpendicular bisector of

AB

passes through the midpoint of the segment and is used as a symmetric fence line. [1]

The farmer uses a rope stretched between

A

and

B

to represent the segment. Explain two ways this method could introduce error into the construction. [1]

The field slopes significantly on one side. Advise the farmer whether the perpendicular bisector method alone is sufficient to guarantee an equal division of the field's actual surface area, justifying your advice. [2]

Solutions

3QuestionConstructing Perpendicular and Angle BisectorsAssessment Practice

6 marks~9 minCriterion C

Landscape designer Zara is planning a garden path that must be equidistant from two fixed points,

A

and

B

, marked on a scale drawing. The line segment

AB

is drawn below.

Construct the perpendicular bisector of

AB

using a compass and straightedge only. All construction arcs must be visible. [2]

Label a point

P

on the perpendicular bisector. Measure and record the distances

PA

and

PB

in centimetres. [1]

Calculate the percentage difference between

PA

and

PB

using:

\text{Percentage Difference} = \frac{|PA - PB|}{\dfrac{PA + PB}{2}} \times 100

[1]

Zara's design specification states the path must be within 5\% of equidistant from

A

and

B

to be approved. Advise Zara whether her construction meets the specification, and explain what any deviation from 0\% reveals about the accuracy of the compass-and-straightedge method. [2]

Solutions

4QuestionConstructing Perpendicular and Angle BisectorsAssessment Practice

6 marks~9 minCriterion B

Points

A(x_1, y_1)

and

B(x_2, y_2)

lie in the Cartesian plane. Their midpoint is

M

and the perpendicular bisector of

AB

passes through

M

.

Four cases are given.

A(0,0),\ B(2,2)

M=(1,1)

, perpendicular bisector

y = -x+2

A(1,2),\ B(3,4)

M=(2,3)

, perpendicular bisector

y = -x+5

A(-1,1),\ B(1,5)

M=(0,3)

, perpendicular bisector

y = -0.5x+3

A(2,-1),\ B(4,3)

M=(3,1)

, perpendicular bisector

y = -0.5x+2.5

Analyse the relationship between the slope of

AB

, the slope of its perpendicular bisector, and the coordinates of

M

in each case. [2]

Deduce a general equation for the perpendicular bisector of

AB

in the form

y = mx + c

, where

m

and

c

are expressed in terms of

x_1,\ y_1,\ x_2,\ y_2

. Show your algebraic reasoning clearly. [3]

A structural engineer checks that a support cable is equidistant from two anchor points

A(3,-2)

and

B(7,4)

. The cable must lie along the perpendicular bisector of

AB

. Apply your formula from part (b) to find the equation of this line, then justify whether the point

(5,1)

lies on the cable's path. [1]

Solutions

Questions 5-37

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