Properties of Triangles 1 Question 2

2. Given, b+c11=c+a12=a+b13 for a ABC with usual notation. If cosAα=cosBβ=cosCγ, then the ordered triad(α,β,γ) has a value

(2019 Main, 11 Jan II)

(a) (19,7,25)

(b) (3,4,5)

(c) (5,12,13)

(d) (7,19,25)

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Answer:

Correct Answer: 2. (d)

Solution:

  1. Given, b+c11=c+a12=a+b13=λ (say)

b+c=11λ,c+a=12λ and a+b=13λ2(a+b+c)=36λa+b+c=18λ

From Eqs. (i) and (ii), we get

a=7λ,b=6λ,c=5λ

Now,

cosA=b2+c2a22bc=λ2[36+2549]60λ2=1260=15cosB=a2+c2b22ac=λ2[49+2536]70λ2=1935 and cosC=a2+b2c22ab=λ2[49+3625]84λ2=6084=57 Thus, cosA=15=735,cosB=1935,cosC=2535cosA7=cosB19=cosC25=135(α,β,γ)=(7,19,25)



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