2026 · Q57
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?
- IThe value of AB is equal to the geometric mean of p and q.
- IIThe perimeter of the triangle is p(q + 1). Select the answer using the code given below:
Elementary MathematicsGeometry: Lines, Angles and Triangles
- AI only
- BII only
- CBoth I and II
- DNeither I nor II
Explanation
By Pythagoras, AB²=AC²−BC²=[(p+q)²−(p−q)²]/4=pq, so AB=√(pq), the geometric mean of p and q — I is correct. The perimeter is p+√(pq), not p(q+1) in general — II is incorrect. Only I is correct, option A.