Question:
In a right-angled triangle Δ ABC , the altitude AB is 5 cm , and the base BC is 12 cm . P and Q are two points on BC such that the areas of ΔABP, ΔABQ , and Δ ABC are in arithmetic progression. If the area of Δ ABC is 1.5 times the area of Δ ABP , the length of PQ , in cm , is
In a right-angled triangle Δ ABC , the altitude AB is 5 cm , and the base BC is 12 cm . P and Q are two points on BC such that the areas of ΔABP, ΔABQ , and Δ ABC are in arithmetic progression. If the area of Δ ABC is 1.5 times the area of Δ ABP , the length of PQ , in cm , is
Updated On: Jan 13, 2026
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The Correct Option is A
Approach Solution - 1
The correct answer is 2.
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Given that ABC is a right-angled triangle with \(AB = 5\) and \(BC = 12\)
=> Area of the triangle = \(0.5 \times 5 \times 12 = 30\). Let us assume \(BP = p, BQ = q\)
=> Area of ABP =\(0.5 \times 5\times p = 2.5p\)
=> Area of ABQ = \(0.5\times5\times q = 2.5q\) Given the area of \(ABC\) is \(1.5\)times that of\(ABP\)
=>\(30 = 1.5 \times2.5p\)
=> \(20 = 2.5p\)
=>\(p = 8.\) Given Areas of \(ABP, ABQ\) and \(ABC\) are in A.P.
=> \(2\times 2.5q = 2.5\times8+ 30\)
=> \(5q = 50\)
=> \(q = 10.\)
\(PQ=BQ - BP = q-p=10-8=2.\)
The correct answer is option \(A \) Answer is 2.
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