Question

In ABC, thelengthofAC=2 Root10. If tanB=2 and tanC=3,find the length of the sides
BC and AB.

Answer 1

Answer:

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Answer 2

In ABC, thelengthofAC=2 Root10. If tanB=2 and tanC=3.

Consider the attached figure while going through the following steps.

Given,

AC = 2√10

tan B = 2

tan C = 3

Construction: Draw a line perpendicular to BC from A and mark it D.

Now consider,

In, Δ ABD,

tan B = 2 = AD/BD

AD = 2BD  ...........(1)

Now consider,

In, Δ ACD,

tan C = 3 = AD/DC

AD = 3DC  .........(2)

comparing (1) and (2), we get,

2BD = 3DC

⇒ BD/CD = 3/2

∴ BD = 3 and CD = 2

BC = 3 + 2

∴ BC = 5

Now consider,

In Δ ACD,

AC² = AD² + CD²

(2√10)² = AD² + 2²

40 = AD² + 4

AD² = 36

AD = 6

Now consider,

In Δ ABD,

AB² = AD² + BD²

AB² = 6² + 3²

AB² = 36 + 9

AB² = 45

AB = 3√5

Hence the lengths of sides BC = 5 and AB = 3√5