Congruence and Similarity

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Congruence and Similarity — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionUsing Congruent Triangles in ProofsConcept Practice

2 marks~3 minCriterion A

A structural engineer uses two triangular steel brackets to reinforce a bridge joint. The brackets are congruent, with

\triangle ABC \cong \triangle DEF

. In

\triangle ABC

, side

AB = 5

cm, side

BC = 7

cm, and

\angle ABC = 64°

Deduce the side in

\triangle DEF

that corresponds to side

AB

, and state its length. [1]

Justify whether the engineer can conclude that

\angle DEF = 64°

without taking any further measurements of

\triangle DEF

. [1]

Solutions

2QuestionCriteria for Similar TrianglesAssessment Practice

6 marks~9 minCriterion D

Surveyors use similar triangles to estimate the heights of tall structures. A surveyor stands on flat ground and measures the angle of elevation to the top of a 3-metre pole as

30°

, and the angle of elevation to the top of a skyscraper as

60°

. The distance from the surveyor to the base of the pole is 10 metres.

Explain how the AA similarity criterion establishes that the two triangles formed are similar, and use this to calculate the height of the skyscraper. [2]

Analyse two sources of error in this method — measurement inaccuracies and terrain irregularities — and explain how each affects the calculated height. [2]

Justify whether the similar-triangles method is sufficiently reliable for this surveying task, or whether additional measurements are required before the result can be used with confidence. [2]

Solutions

3QuestionUsing Scale Factor in Similar FiguresAssessment Practice

6 marks~9 minCriterion B

A designer produces a series of similar pentagonal prisms for architectural models. The table below summarises measurements for the first three prisms in the series.

Side length (cm)	2	4	6
Perimeter of base (cm)	10	20	30
Area of base (cm²)	6.88	27.52	61.92
Volume (cm³)	34.40	275.20	928.80

In each prism, the height equals the side length.

Deduce a formula for the perimeter

P

of the pentagonal base in terms of its side length

s

. [1]

Deduce a formula for the area

A

of the pentagonal base in terms of

s

, showing that the ratio

\dfrac{A}{s^2}

is constant. [2]

Using your results from (a) and (b), deduce a formula for the volume

V

of the prism in terms of

s

. [1]

A client requests a prism with side length 8 cm, but the display cabinet has a maximum volume of 850 cm³. Calculate the volume of this prism and advise the client whether the prism can be displayed in the cabinet, justifying your answer. [2]

Solutions

4QuestionCriteria for Triangle Congruence SSS SAS ASA RHSAssessment Practice

6 marks~9 minCriterion C

A structural engineer is checking whether two triangular roof trusses,

\triangle ABC

and

\triangle DEF

, are identical in shape and size before installation. She records measurements progressively.

She finds

AB = DE

and

BC = EF

. Explain why this information alone is insufficient to confirm the trusses are congruent. [2]

She then confirms

\angle ABC = \angle DEF

. Justify why

\triangle ABC \cong \triangle DEF

can now be concluded, naming the congruence criterion used. [2]

A colleague suggests that knowing only

\angle ABC = \angle DEF

and

BC = EF

(without

AB = DE

) would be equally sufficient. Advise the engineer whether this measurement combination is reliable for confirming congruence, and state what additional measurement she would need. [2]

Solutions

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