माना, Δ ABC ~ Δ QPR है और \(\rm \frac{area \ of \ \Delta ABC}{area \ of \ \Delta PQR}=\frac{64}{25}\) है। यदि QP = 12 सेमी, PR = 10 सेमी और AC = 15 सेमी है, तो AB की लंबाई ज्ञात कीजिए:

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AAI Junior Assistant (Fire Service) Official Paper (Held On: 15 Nov 2022 Shift 1)
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  1. 12.8 सेमी
  2. 24 सेमी
  3. 16 सेमी
  4. 19.2 सेमी

Answer (Detailed Solution Below)

Option 4 : 19.2 सेमी
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दिया गया है:

Δ ABC ~ Δ QPR

\(\frac{area \ of \ Δ ABC}{area \ of \ Δ PQR}=\frac{64}{25}\)

प्रयुक्त सूत्र:

समान त्रिभुजों में, क्षेत्रफलों का अनुपात संगत भुजाओं के अनुपात के वर्ग के बराबर होता है।

\(\frac{area \ of \ Δ ABC}{area \ of \ Δ PQR} = (\frac{corresponding \ side \ of \ Δ ABC}{corresponding \ side \ of \ Δ PQR})^2\)

गणना:

5-5-2025 IMG-1083 Prakhar -2

\(\frac{64}{25} = (\frac{AB}{QP})^2\)

\(\frac{64}{25} = (\frac{AB}{12})^2\)

\(\frac{64}{25} = \frac{AB^2}{144}\)

\(\frac{64}{25} × 144 = AB^2\)

\(\frac{64 × 144}{25} = AB^2\)

\(\frac{9216}{25} = AB^2\)

⇒ 368.64 = \(AB^2\)

⇒ AB = \(\sqrt{368.64}\)

⇒ AB = 19.2 सेमी

∴ सही उत्तर विकल्प (4) है।

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