Question

If the angles of a triangle are in the ratio \[1:2:3\], then what is the ratio of their corresponding sides?
A. \[1:2:3\]
B. \[1:\sqrt 3 :2\]
C. \[\sqrt 2 :\sqrt 3 :2\]
D. \[1:\sqrt 3 :3\]

Answer 1

Hint: First, using the formula of the sum of the angles of a triangle calculate the measurement of each angle. Then, check the type of the triangle. After that, calculate the ratio of the corresponding sides to get the required answer.

Formula used:
Trigonometric values of an angle:
\[\sin 30^ {\circ } = \dfrac{1}{2}\]
\[\sin 60^ {\circ } = \dfrac{{\sqrt 3 }}{2}\]
\[\sin 90^ {\circ } = 1\]

Complete step by step solution:

Calculate the measurement of the angles:
Given, the angles are in the ratio \[1:2:3\]
Let \[x\] be the common multiple.
Then, by angle sum property of a triangle.
\[1x + 2x + 3x = 180^ {\circ }\]
\[ \Rightarrow 6x = 180^ {\circ }\]
\[ \Rightarrow x = 30^ {\circ }\]

Therefore, the angles of the triangles are \[30^ {\circ },60^ {\circ }\] and \[90^ {\circ }\].
Here, the triangle is a right-angle triangle.

Hence, the corresponding sides are in the ratio as follows:
\[\sin 30^ {\circ }:\sin 60^ {\circ }:\sin 90^ {\circ }\]
\[ \Rightarrow \dfrac{1}{2}:\dfrac{{\sqrt 3 }}{2}:1\]
Multiply each value by 2.
\[ \therefore 1:\sqrt 3 :2\]
Hence the correct option is B.

Note:
Trigonometric ratios are the ratios of sides of a right-angle triangle. The mostly used trigonometric ratios are sine, cosine, and tangent.
The sine of an angle is the ratio of the opposite side to that angle to the hypotenuse.
The sum of all interior angles of any triangle is \[180^ {\circ }\].