CDS 2026 · PYQ · Geometry / Right Triangle · medium
ABC is a triangle right-angled at B. If AC = (p + q)/2 and BC = (p − q)/2, then which of the following statements is/are correct?
I. The value of AB is equal to the geometric mean of p and q.
II. The perimeter of the triangle is p(q + 1).
A.I only✓ Correct
B.II only
C.Both I and II
D.Neither I nor II
Explanation
By Pythagoras: AB² = AC² − BC² = [(p+q)/2]² − [(p−q)/2]² = [(p+q)² − (p−q)²]/4 = (4pq)/4 = pq. So AB = √(pq), the geometric mean of p and q. I is correct. For II: perimeter = AB + BC + AC = √(pq) + (p−q)/2 + (p+q)/2 = √(pq) + p. This is not equal to p(q + 1) = pq + p in general. So II is incorrect. Only I is correct.
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