Question

If angles of a triangle are in the ratio of \[2:3:7\] then the sides are in the ratio of
A. \[\sqrt 2 :2:(\sqrt 3 + 1)\]
B. \[2:\sqrt 2 :(\sqrt 3 + 1)\]
C. \[\sqrt 2 :(\sqrt 3 + 1):2\]
D. \[2:(\sqrt 3 + 1):\sqrt 2 \]

Answer 1

Hints:
In this case, we have been given the triangle’s angle ratio as \[2:3:7\] and we are to find the ratio of their sides. For that we have to write the angle’s ratio in terms of sum of angles of triangles. As we already know that the angle’s sum will be \[{180^\circ }\] and now we have to equate each term to \[{180^\circ }\] for all angles. Then we have should value the angles and hence the ratio of sides.
Formula used:
Formula used here is angle sum property
Triangle’s sum of all angles is \[{180^\circ }\]
Complete Step-by Step Solution:
Given that the triangle's angles are in the ratio of 
\[2:3:7\]
We have been already known that the triangle’s sum of angles is
\[{180^\circ }\]
Therefore the equation can be written according to the property,
\[2x + 3x + 7x = 180\]---- (1)
Now, we have to group the like terms and add we get
\[ \Rightarrow 12{\rm{x}} = 180\]
On solving the above obtained equation we get
\[ \Rightarrow x = 15\]
Therefore, we can determine the first angle by substituting the value \[x = 15\] in equation (1)
\[2{\rm{x}} = 2*15 = {30^\circ }\]
Now, let us determine the second angle by same method as above, we get
\[3{\rm{x}} = 3*15 = {45^\circ }\]
Now, to determine the third angle same method should be followed we get
\[7{\rm{x}} = 7*15 = {105^\circ }\]
Now, we have obtained the answers. Let us arrange the sides in ratio form, we get
\[\sin 30:\sin 45:\sin 105\]
By using trigonometry ratios, we obtained the values as
\[ = \frac{1}{2}:\frac{1}{{\sqrt 2 }}:\frac{{\sqrt 3 + 1}}{{2\sqrt 2 }}\]
Now, we have to simplify further, we obtain
\[ = \sqrt 2 :2:\sqrt 3 + 1\]
Therefore, the sides are in the ratio of \[\sqrt 2 :2:(\sqrt 3 + 1)\]
Hence, the option A is correct
Note:
The angle sum property asserts that the sum of a triangle's internal angles is always 180. Student should understand that if one of the triangle's angles is 90 degrees, the other two will be acute angles. And a triangle with a 90-degree angle is known as a right-angled triangle.