CDS 2026 (I) Elementary Mathematics, question 74

Topic: Geometry

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Q74

A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.

Question : P, Q, R and S are respectively the midpoints of sides AB, BC, CD, DA of a rhombus ABCD. These points are joined to form a quadrilateral PQRS. Is the quadrilateral cyclic ?

Statement-I : ∠ABC=120∘\angle ABC=120^{\circ}

Statement-II : ∠BCD=60∘\angle BCD=60^{\circ}

Which one of the following is correct in respect of the above Question and the Statements ?

  1. (a) The Question can be answered by using one of the statements alone, but cannot be answered using the other statement alone
  2. (b) The Question can be answered by using either Statement alone
  3. (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
  4. (d) The Question can be answered even without using both the StatementsCorrect answer

Solution.

Answer: (d) The Question can be answered even without using both the Statements

  1. The midpoint theorem gives PQ∥ACPQ \parallel AC and QR∥BDQR \parallel BD, with the other sides parallel to these.
  2. The diagonals of a rhombus are perpendicular, so PQRS is a rectangle.
  3. Every rectangle is cyclic because its opposite angles sum to 180°.
  4. No particular angle of the rhombus is needed.