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Consider the following statements concerning a triangle ABC (i) The sides a, b, c and area ∆ are rational. (ii) a, tan B/2, tan C/2 are rational (iii) a, sin A, sin B, sin C are rational. Prove that (i) ⇒ (ii) ⇒ (iii) ⇒ (i)

Consider the following statements concerning a triangle ABC
(i) The sides a, b, c and area ∆ are rational.
(ii) a, tan B/2, tan C/2 are rational
(iii) a, sin A, sin B, sin C are rational.
Prove that (i) ⇒ (ii) ⇒ (iii) ⇒ (i)

Grade:11

1 Answers

Jitender Pal
askIITians Faculty 365 Points
7 years ago
Hello Student,
Please find the answer to your question
(I) a, b, c and ∆ are rational.
⇒ s = a + b + c/2 is also rational
⇒ tan B/2 = √(s – a) (s – c)/s (s – b) = ∆/s (s – b) is also rational
And tan C/2 = √(s – a) (s – b)/s (s – c) = ∆/s (s – c) is also rational
Hence ( I) ⇒ (II).
(II) a, tan B/2, tan C/2 are rational.
⇒ sin B = 2tan B/2 / a + tan2 B/2
And sin C = 2 tan C/2 / 1 + tan2 C/2 are rational
And tan A/2 = tan [90° - (B/2 + C/2)] = cot (B/2 + C/2)
= 1/tan (B/2 + C/2) = 1 – tan B/2 tan C/2 / tan B/2 + tan C/2 is rational
∴ sin A = 2tan A/2 / 1 + tan2 A/2 is rational.
Hence (II) ⇒ (III)
(III) a, sin A, sin B, sin C are rational.
But a/sin A = 2R
⇒ R is rational
∴ b = 2R sin B, c = 2R sin C are rational.
∴ ∆ = 1/2 bc sin A is rational
Hence (III) ⇒ (I).

Thanks
Jitender Pal
askIITians Faculty

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