Multiple choice

I is a point inside the triangle ABC, the bisectors of ∠BIC, ∠AIC and ∠AIB meet the sides BC, CA and AB in points E, F, and D respectively. The lengths (in cm) of AD, DB, AF, FC and BE are 3, 5, 4, 4, 6 respectively. What is the length of CE? त्रिभुज ABC के अन्दर एक बिन्दु I है। ∠BIC, ∠AIC और∠AIB के समद्विभाजक भुजा BC, CA और AB को क्रमशः बिन्दु E, F और D पर मिलते है। AD, DB, AF, FC और BE की लम्बाईयां क्रमशः 3, 5, 4, 4 और 6 है। तो CE की लम्बाई क्या है?

  1. 2 cm/सेमी

  2. 3.6 cm/सेमी

  3. 4.4 cm/सेमी

  4. None of these/इनमें से कोई नहीं
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B Correct answer
Explanation

By Ceva's Theorem for concurrent cevians: (AD/DB) × (BE/EC) × (CF/FA) = 1. Given AD=3, DB=5, BE=6, AF=4, FC=4. So (3/5) × (6/CE) × (4/4) = 1. Thus (18/5CE) = 1, giving CE = 18/5 = 3.6 cm. The angle bisectors of the triangle formed by I and vertices are concurrent at I.

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