Congruence and Similarity

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Congruence and Similarity — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionCriteria for Triangle Congruence SSS SAS ASA RHSAssessment Practice

2 marks~3 minCriterion D

Prefabricated roof trusses are manufactured to precise dimensions so that the SSS congruence criterion can confirm all trusses are identical, ensuring uniform load distribution across a structure.

Discuss one limitation of relying on the SSS congruence criterion to verify that manufactured roof trusses are congruent, and justify whether this limitation poses a genuine risk to structural safety after installation. [2]

Solutions

2QuestionReal-Life Applications of Similarity and CongruenceAssessment Practice

4 marks~6 minCriterion A

A surveyor needs to find the height of a tall building. She stands a pole of height 3 m vertically near the building. The pole casts a shadow of 2 m, and the building casts a shadow of 30 m. The sun's rays strike both objects at the same angle, forming two similar triangles.

Construct a clearly labelled diagram showing the two similar triangles formed by the pole, the building, and their shadows. Label all known lengths and indicate the equal angles. [1]

Deduce a proportion relating the height of the pole, the length of the pole's shadow, the height of the building

h

, and the length of the building's shadow. [1]

Solve your proportion from part (b) to find

h

. Show all working. [1]

The building's shadow measurement carries an uncertainty of

\pm 0.5

m. Advise the surveyor whether this shadow measurement is precise enough to report the building's height reliably for a construction project requiring accuracy to the nearest metre. [1]

Solutions

3QuestionCriteria 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

4QuestionUsing 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

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