NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Congruence and similarity

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A school in Brisbane is ordering triangular metal brackets for a physics experiment. Each bracket must be identical in shape and size.

Bracket ABC has side lengths AB = 33 cm, BC = 77 cm, and AC = 55 cm. Bracket DEF has side lengths DE = 33 cm, EF = 77 cm, and DF = 55 cm.

A transformation sequence consisting of a rotation followed by a reflection maps triangle DEF exactly onto triangle ABC. Using the conditions for triangle congruence, determine which congruence test confirms that triangle ABC is congruent to triangle DEF.

Choose the condition directly established by the three stated corresponding side equalities. Answer as a single exact value: SSS, SAS, ASA, or RHS.

Triangle DEF side measurementsDE = 33 cm; DF = 55 cm; EF = 77 cmDEF33 cm55 cm77 cm
Measurements from the question · not to scale.
Triangle ABC side measurementsAB = 33 cm; AC = 55 cm; BC = 77 cmABC33 cm55 cm77 cm
Measurements from the question · not to scale.
Concept illustrations

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Corresponding lengths scale together. The right triangles have side lengths 3,4,5 and 9,12,15. The length scale factor is 3 and the area scale factor is 9. Both use the same drawing scale.34912155×3
The right triangles have side lengths 3,4,5 and 9,12,15. The length scale factor is 3 and the area scale factor is 9. Both use the same drawing scale.
Coordinates on a plane. Illustration: the point (2, 2) lies on y = x. Read x across, then y up.-2-2-1-111223344xy
Illustration: the point (2, 2) lies on y = x. Read x across, then y up.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. Given: Triangle ABC has side lengths AB = 33 cm, BC = 77 cm, and AC = 55 cm.
  2. Given: Triangle DEF has side lengths DE = 33 cm, EF = 77 cm, and DF = 55 cm.
  3. Identify corresponding sides: AB corresponds to DE (both 33 cm), BC corresponds to EF (both 77 cm), and AC corresponds to DF (both 55 cm).
  4. Three pairs of corresponding sides are equal: AB = DE, BC = EF, and AC = DF.
  5. The condition for triangle congruence based on three equal corresponding sides is the SSS (Side-Side-Side) congruence rule.
  6. The SSS rule states that if all three sides of one triangle are equal in length to the corresponding sides of another triangle, then the triangles are congruent.
  7. The transformation sequence (rotation followed by reflection) confirms that one triangle can be mapped exactly onto the other, which is consistent with congruence.
  8. Since all three corresponding sides are equal, and no angles are given, the only congruence test directly established by the side equalities is SSS.
  9. SAS (Side-Angle-Side) would require two sides and the included angle, but no angle information is provided.
  10. ASA (Angle-Side-Angle) requires two angles and the included side, which is not given.
  11. RHS (Right angle-Hypotenuse-Side) applies only to right-angled triangles, and no right angle is indicated.
  12. Therefore, the congruence test directly established by the three stated corresponding side equalities is SSS.

Reference answer: SSS

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