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 [katex]\triangle ABC[/katex] with the side [katex]a[/katex], [katex]b[/katex] and [katex]c[/katex] and angle [katex]\alpha[/katex], [katex]\beta[/katex] and [katex]\gamma[/katex].We can observe that we are given the three sides [katex]a[/katex], [katex]b[/katex] and [katex]c[/katex]. Therefore the figure illustrates a triangle combination which is known as a SSS triangle.

How to Solve 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:

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

Example No.1

(By using Law of cosine)

when three sides are given.

a=7 , b=3 , c=5

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

By putting the values

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

[katex]\Rightarrow\cos\alpha=\frac{9+25-49}{30}[/katex].

[katex]\Rightarrow\cos\alpha=\frac{-15}{30}[/katex].

[katex]\Rightarrow\cos\alpha=\frac{-1}{2}[/katex].

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

[katex]\Rightarrow\boxed{\alpha\;=120^o}[/katex].

Now,

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

using values

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

[katex]\Rightarrow\cos\beta=\frac{25+49-9}{70}[/katex].

[katex]\Rightarrow\cos\beta=\frac{65}{70}[/katex].

[katex]\Rightarrow\cos\beta=0.9286[/katex].

[katex]\Rightarrow\beta\;\;\;=\cos^{-1}(0.9286)[/katex].

[katex]\Rightarrow\boxed{\beta\;=1^o}[/katex].

Now, we know that

[katex]\alpha +\beta+ \gamma=180^o[/katex].

[katex] \gamma=180^o-\alpha -\beta[/katex].

[katex] \gamma=180^o-120^o -1^o[/katex].

[katex] \Rightarrow\boxed{\gamma=59^o}[/katex].

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