CDS 2026 (I) Elementary Mathematics, question 59

Topic: Geometry

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Q59

ABCD is a quadrilateral with sides AB=9AB=9 cm, BC=40BC=40 cm, CD=28CD=28 cm and DA=15DA=15 cm and one of the diagonals AC=41AC=41 cm. Which of the following statements is/are correct ?

I. The vertices A, B, C and D of the quadrilateral lie on a circle.

II. The area of triangle ACD is 126 cm2126\text{ cm}^{2}.

Select the answer using the code given below :

  1. (a) I only
  2. (b) II onlyCorrect answer
  3. (c) Both I and II
  4. (d) Neither I nor II

Solution.

Answer: (b) II only

  1. Since 92+402=412,∠ABC=90∘9^{2} + 40^{2} = 41^{2},\angle ABC = 90^{\circ}.
  2. For ABCD to be cyclic, the opposite angle ADC would also need to be 90°; but 152+282=1009≠41215^{2} + 28^{2} = 1009 \neq 41^{2}.
  3. Thus I is false.
  4. For triangle ACD, the semiperimeter is (15+28+41)/2=42(15 + 28 + 41)/2 = 42.
  5. Heron's formula gives area =42×27×14×1=126cm2= \sqrt{42 \times 27 \times 14 \times 1} = 126{cm}^{2}, proving II.