Question

Points D and E are marked on sides AC and BC, respectively. It is known that AB = BD, ∠ABD = 56∘, ∠DEC = 90∘. Find ∠BDE if it is known that 2DE = AD.

Answer 1

Given : Points D and E are marked on sides AC and BC AB = BD, ∠ABD = 56∘, ∠DEC = 90∘ & 2DE = AD.

To find : ∠BDE

Solution:

in Δ ABD

AB = BD   &  ∠ABD = 56°

=> AD² = AB² + BD² - 2AB.BDCos56°

=> AD² = BD² + BD² - 2BD²Cos56°

=> AD² = 2BD² (1  - Cos56°)

=>  AD² = 2BD² (2Sin²28°)

=> AD = 2BDSin28°

2DE  = AD

=> 2DE = 2BDSin28°

=> DE/BD = Sin28°

∠DEC = 90°  => ∠DEB = 90°  as E is on BC

in Δ DEB

Sin∠DBE  =  DE/BD

=> Sin∠DBE  = Sin28°

=> ∠DBE  = 28°

∠BDE + ∠DBE   + ∠DEB  = 180°

=> ∠BDE + 28° + 90° = 180°

=> ∠BDE = 62°

∠BDE = 62°

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