Construction and Loci

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Construction and Loci — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionConstructing Triangles with Given ConditionsAssessment Practice

4 marks~6 minCriterion C

A city planner is repositioning a triangular green space. On a coordinate grid, triangle

ABC

has vertices

A(1, 2)

B(4, 2)

, and

C(3, 5)

. After repositioning, triangle

A'B'C'

has vertices

A'(4, 6)

B'(7, 6)

, and

C'(6, 9)

Calculate the translation vector

\mathbf{v} = \begin{pmatrix} \Delta x \\ \Delta y \end{pmatrix}

that maps

A

onto

A'

. [1]

Verify that

\mathbf{v}

also maps

B

onto

B'

and

C

onto

C'

. [1]

The planner states: "The repositioned green space covers a completely different area of the city." Justify whether this statement is mathematically supported. [2]

Solutions

2QuestionConstructing Triangles with Given ConditionsAssessment Practice

6 marks~9 minCriterion D

Surveyors use triangulation to map a triangular field. Two sides measure

150 \text{ m}

and

200 \text{ m}

, with an included angle of

60°

. The surveying instrument has a measurement uncertainty of

\pm 0.5°

Calculate the area of the triangular field using the measured values. [2]

Determine the maximum and minimum possible areas of the field, given the uncertainty in the angle measurement. Show all working. [2]

The surveying company must report the field area to within

\pm 100 \text{ m}^2

of the true value to satisfy a legal land registration requirement. Advise the surveying company whether the instrument is sufficiently precise for this purpose, justifying your answer with reference to your results from parts (a) and (b). [2]

Solutions

3QuestionShading Regions Using Multiple LociAssessment Practice

6 marks~9 minCriterion A

A rectangular field is used for a treasure hunt. The treasure's location is determined by distance constraints from three landmarks: a tree at point

A

, a rock at point

B

located

8

m from

A

, and a straight fence running along one edge of the field.

Construct the locus of points exactly

5

m from

A

, and shade the region within

5

m of

A

. [1]

The treasure is also within

4

m of point

B

. Construct both circular loci on the same diagram and shade the region satisfying both constraints simultaneously. [2]

The treasure must additionally be within

3

m of the fence. Add this locus to your diagram, shade the region satisfying all three constraints, and justify whether the three constraints are sufficient to uniquely locate the treasure. [3]

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

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