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0580/42 Oct/Nov 2023, Question 10: Congruent Triangles: Proof Conditions

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 7 marks

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The question, in short

Complete congruence statements for a regular hexagon with diagonals DF, DA, DB, and for two triangles formed by tangents TP, TQ from T to a circle centre O.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 2 marks

Using three different triangles, name the triangle congruent to DEF and another triangle congruent to a named one.

  1. 1DE = BC, EF = CD and angle DEF = angle BCD = 120°: triangle DEF is congruent to triangle BCD
  2. 2DF = DB, AF = AB, AD common: triangle ADF is congruent to triangle ADB

AnswerTriangle DEF is congruent to triangle BCD; Triangle ADF is congruent to triangle ADB

Part (b) · 5 marks

Fill in equal sides, the 90° angle, its reason, the congruence criterion, and what this proves about tangents TP and TQ.

  1. 1OP = OQ because each is a radius of the circle
  2. 2Angle OPT = angle OQT = 90° because a tangent is perpendicular to the radius
  3. 3Right angle, hypotenuse OT, side OP = OQ: congruent by RHS
  4. 4So TP = TQ: the tangents are equal

AnswerOQ; OQT; tangent perpendicular to radius; RHS; equal

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Also searched as: 0580/42/O/N/23 · 0580_w23_qp_42 · 0580 w23 42 · question 10.
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