\begin{array} { l } { \text { If in } \Delta A B C \text { the ratio o

1.	\begin{array} { l } { \text { If in } \Delta A B C \text { the ratio of the angles is } 1 : 2 : 3 , \text { then } } \\ { \text { prove that } a : c = 1 : \sqrt { 3 } : 2 } \end{array}
Answer» Let the angles be k, 2k and 3k. Then, k + 2k + 3k = 180° ⇒ 6k = 180° ⇒ k = 30° So, the angles are 30°, 60° and 90° Let x, y, and z denote the sides opposite to these angles. Then, x/sin 30° = y/sin 60° = c/sin 90° ⇒ x : y : z = sin 30° : sin 60° : sin 90° ⇒ x : y : z = ½ : √3/2 : 1 ⇒ x : y : z = 1 : √3 : 2.

\begin{array} { l } { \text { If in } \Delta A B C \text { the ratio of the angles is } 1 : 2 : 3 , \text { then } } \\ { \text { prove that } a : c = 1 : \sqrt { 3 } : 2 } \end{array}

Answer»

Let the angles be k, 2k and 3k.

Then, k + 2k + 3k = 180°

⇒ 6k = 180°

⇒ k = 30°

So, the angles are 30°, 60° and 90°

Let x, y, and z denote the sides opposite to these angles.

Then, x/sin 30° = y/sin 60° = c/sin 90°

⇒ x : y : z = sin 30° : sin 60° : sin 90°

⇒ x : y : z = ½ : √3/2 : 1

⇒ x : y : z = 1 : √3 : 2.