CDS 2026 (I) Elementary Mathematics, question 57

Topic: Geometry

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Q57

ABC is a triangle right-angled at B. If AC=(p+q)2AC=\frac{(p+q)}{2} and BC=(p−q)2BC=\frac{(p-q)}{2}, then which of the following statements is/are correct ?

I. The value of AB is equal to the geometric mean of pp and qq.

II. The perimeter of the triangle is p(q+1)p(q+1).

Select the answer using the code given below :

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

Solution.

Answer: (a) I only

  1. Pythagoras gives AB2=[(p+q)2−(p−q)2]/4=pqAB^{2} = \left\lbrack (p + q)^{2} - (p - q)^{2} \right\rbrack/4 = pq.
  2. Thus AB=pqAB = \sqrt{pq}, proving I.
  3. The perimeter is (p+q)/2+(p−q)/2+pq=p+pq(p + q)/2 + (p - q)/2 + \sqrt{pq} = p + \sqrt{pq}, not p(q+1)p(q + 1) in general.