# How to Solve SSS Triangle

“SSS” means side, side, side.

“SSS” is used when three sides of a triangle are given and we want to find the missing angles. Consider the triangle \triangle ABC with the side a, b and c and angle \alpha, \beta and \gamma.We can observe that we are given the three sides a, b and c. Therefore the figure illustrates a triangle combination which is known as a SSS triangle.

To solve an SSS triangle:

Step 1: First we use the “Law of cosine ” to find an angle.

Step 2: Secondly use again “Law of cosine ” to find another angle.

Step 3: Finally use angles of triangles added to 180 to find the last angle.

We use the “angle” version of the Law of Cosines:

• \cos\alpha=\frac{b^2+c^2-a^2}{2bc}.
• \cos\beta=\frac{a^2+c^2-b^2}{2ac}.
• \cos\gamma=\frac{a^2+b^2-c^2}{2ab}.

Example No.1

(By using Law of cosine)

when three sides are given.

a=7 , b=3 , c=5

\cos\alpha=\frac{b^2+c^2-a^2}{2bc}.

By putting the values

\cos\alpha=\frac{3^2+5^2-7^2}{2(3)(5)}.

\Rightarrow\cos\alpha=\frac{9+25-49}{30}.

\Rightarrow\cos\alpha=\frac{-15}{30}.

\Rightarrow\cos\alpha=\frac{-1}{2}.

\Rightarrow\alpha\;\;\;=\cos^{-1}(\frac{-1}2).

\Rightarrow\boxed{\alpha\;=120^o}.

Now,

\cos\beta=\frac{c^2+a^2-b^2}{2ca}.

using values

\cos\beta=\frac{5^2+7^2-3^2}{2(5)(7)}.

\Rightarrow\cos\beta=\frac{25+49-9}{70}.

\Rightarrow\cos\beta=\frac{65}{70}.

\Rightarrow\cos\beta=0.9286.

\Rightarrow\beta\;\;\;=\cos^{-1}(0.9286).

\Rightarrow\boxed{\beta\;=1^o}.

Now, we know that

\alpha +\beta+ \gamma=180^o.

\gamma=180^o-\alpha -\beta.

\gamma=180^o-120^o -1^o.

\Rightarrow\boxed{\gamma=59^o}. 