Congruence and Similarity
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Congruence and Similarity — Free MYP4 Mathematics (Standard) 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 △ABC≅△DEF\triangle ABC \cong \triangle DEF. In △ABC\triangle ABC, side AB=5AB = 5 cm, side BC=7BC = 7 cm, and ∠ABC=64°\angle ABC = 64°.
a
Deduce the side in △DEF\triangle DEF that corresponds to side ABAB, and state its length. [1]
b
Justify whether the engineer can conclude that ∠DEF=64°\angle DEF = 64° without taking any further measurements of △DEF\triangle DEF. [1]
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2QuestionUsing Scale Factor in Similar FiguresAssessment Practice
4 marks~6 minCriterion D
A group of hikers plans a route using a topographic map with a scale of 1:50 0001 : 50\,000.
a
The distance between starting point AA and campsite BB measures 8.58.5 cm on the map. Calculate the actual distance in kilometres between AA and BB. [1]
b
The total elevation gain along the route from AA to BB is 400400 m. Show that the average slope of the terrain, expressed as a percentage, is approximately 9.4%9.4\%. [1]
c
A hiking guidebook classifies any route with an average slope above 10%10\% as strenuous. Using your results from parts (a) and (b), advise the group of hikers whether the route from AA to BB should be classified as strenuous, and explain one real-world factor that could make the actual difficulty differ from this classification. [2]
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3QuestionCriteria for Triangle Congruence SSS SAS ASA RHSAssessment Practice
6 marks~9 minCriterion C
A structural engineer is checking whether two triangular roof trusses, △ABC\triangle ABC and △DEF\triangle DEF, are identical in shape and size before installation. She records measurements progressively.
a
She finds AB=DEAB = DE and BC=EFBC = EF. Explain why this information alone is insufficient to confirm the trusses are congruent. [2]
b
She then confirms ∠ABC=∠DEF\angle ABC = \angle DEF. Justify why △ABC≅△DEF\triangle ABC \cong \triangle DEF can now be concluded, naming the congruence criterion used. [2]
c
A colleague suggests that knowing only ∠ABC=∠DEF\angle ABC = \angle DEF and BC=EFBC = EF (without AB=DEAB = 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]
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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)246
Perimeter of base (cm)102030
Area of base (cm²)6.8827.5261.92
Volume (cm³)34.40275.20928.80


In each prism, the height equals the side length.
a
Deduce a formula for the perimeter PP of the pentagonal base in terms of its side length ss. [1]
b
Deduce a formula for the area AA of the pentagonal base in terms of ss, showing that the ratio As2\dfrac{A}{s^2} is constant. [2]
c
Using your results from (a) and (b), deduce a formula for the volume VV of the prism in terms of ss. [1]
d
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]
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