In a Δ ABC, following relations hold good. In which case(s) the triangle is a right angled triangle?
Text Solution
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(a,b,c,d)
+ 
= 
– 
⇒ 
= 
⇒ tan 2 
= 1 ⇒ A = 90°
4R 2 (sin 2 A + sin 2 B + sin 2 C) = 8R 2
1 – (cos 2 A – sin 2 B) + sin 2 C = 2
1 – cos(A + B) cos(A – B) + 1 – cos 2 C = 2.
cosC cos(A – B) – cos 2 C = 0
cosC[cos(A – B) – cosC] = 0
cosC[cos(A – B) + cos (A + B)] = 0
2 cosA cosB cosC = 0
⇒ A = 90° or B = 90° or C = 90°
r 1 = s.
s tan A/2 = s ⇒ tan A/2 = 1 ⇒ A = 90°
= 
⇒ 
= tanA/2
⇒ 2 sin 2 A/2 = 1
⇒ 1 – cosA = 1 ⇒ cosA = 0 ⇒ A = 90°
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