The ratio of lengths of opposite sides of 30°, 60°, 90° isA) 1 : 2

1.	The ratio of lengths of opposite sides of 30°, 60°, 90° isA) 1 : 2 : 3 B) 1 : 1 : √2 C) 1 : 3 : 2 D) 1 : √3 : 2
Answer» Correct option is: D) 1 : √3 : 2 By sine formula for triangle, \(\frac a {sin\, A} = \frac b{sin\,B} = \frac c{sin \, C} = \lambda\) (Let) = a = \(\lambda\) sin A, where a is length of side opposite to <A. b = \(\lambda\) sin B, where b is length of side opposite to < B. c = \(\lambda\) sin C, where c is length of side opposite to < C. Let A = \(30^\circ\), B = \(60^\circ\) & C = \(90^\circ\) Then a = \(\lambda\) sin \(30^\circ\) = \(\frac \lambda2\) b = \(\lambda\) sin \(60^\circ\) = \(\frac {\sqrt3 \lambda}2\) c = \(\lambda\) sin \(90^\circ\) = \(\lambda\) \(\therefore\) a : b : c = \(\frac \lambda2\) : \(\frac {\sqrt3 \lambda}2\) : \(\lambda\) = \(\lambda\) : \(\sqrt3 \lambda\) : 2\(\lambda\) (\(\because\) On multiplying by 2) = 1 : \(\sqrt3\) : 2 (\(\because\) On multiplying by \(\lambda\)) Hence, the ratio of lengths of opposite side of \(30^\circ\), \(60^\circ\) , \(90^\circ\) is 1 : \(\sqrt3\): 2 Correct option is: D) 1 : √3 : 2

The ratio of lengths of opposite sides of 30°, 60°, 90° isA) 1 : 2 : 3 B) 1 : 1 : √2 C) 1 : 3 : 2 D) 1 : √3 : 2

Answer»

Correct option is: D) 1 : √3 : 2

By sine formula for triangle,

\(\frac a {sin\, A} = \frac b{sin\,B} = \frac c{sin \, C} = \lambda\) (Let)

= a = \(\lambda\) sin A, where a is length of side opposite to <A.

b = \(\lambda\) sin B, where b is length of side opposite to < B.

c = \(\lambda\) sin C, where c is length of side opposite to < C.

Let A = \(30^\circ\), B = \(60^\circ\) & C = \(90^\circ\)

Then a = \(\lambda\) sin \(30^\circ\) = \(\frac \lambda2\)

b = \(\lambda\) sin \(60^\circ\) = \(\frac {\sqrt3 \lambda}2\)

c = \(\lambda\) sin \(90^\circ\) = \(\lambda\)

\(\therefore\) a : b : c = \(\frac \lambda2\) : \(\frac {\sqrt3 \lambda}2\) : \(\lambda\)

= \(\lambda\) : \(\sqrt3 \lambda\) : 2\(\lambda\) (\(\because\) On multiplying by 2)

= 1 : \(\sqrt3\) : 2 (\(\because\) On multiplying by \(\lambda\))

Hence, the ratio of lengths of opposite side of \(30^\circ\), \(60^\circ\) , \(90^\circ\) is 1 : \(\sqrt3\): 2

Correct option is: D) 1 : √3 : 2