“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}.