How to find the missing length of an obtuse triangle if two sides have a length of 10 one of the angles is 120 degrees?

Hint: In this question we need to find the length of side of an obtuse triangle, for this we need to know the other two sides and the opposite angle, we will need to use the version of the Cosine Rule where ${a^2}$ is the subject of the formula, which is given by,
${a^2} = {b^2} + {c^2} - 2bc\cos A$, Side $a$ is the one we are trying to find. Sides $b$ and $c$ are the other two sides, and angle $A$ is the angle opposite side $a$, by substituting the given values in the formula we will get the required length of the side.

Complete step by step answer:
Given two sides have a length of 10 one of the angles is 120 degrees,
So, here angle = ${120^o}$, and
Length of the two sides which are equal = 10,
So, we need to find the length of the other side, we will use the version of the Cosine Rule where ${a^2}$ is the subject of the formula, which is given by,
${a^2} = {b^2} + {c^2} - 2bc\cos A$, Side $a$ is the one we are trying to find. Sides $b$and $c$are the other two sides, and angle $A$is the angle opposite side $a$.
So, here $b = c = 10$, and $A = {120^o}$, and we need to find the side $a$, by substituting the values in the formula we get,
$ \Rightarrow {a^2} = {b^2} + {c^2} - 2bc\cos A$,
Now substituting the values we get,
$ \Rightarrow {a^2} = {10^2} + {10^2} - 2\left( {10} \right)\left( {10} \right)\cos {120^o}$,
We know that $\cos {120^o} = \dfrac{{ - 1}}{2}$, by substituting the values we get,
$ \Rightarrow {a^2} = 100 + 100 - 2\left( {10} \right)\left( {10} \right)\cos \left( {\dfrac{{ - 1}}{2}} \right)$,
Now simplifying we get,
$ \Rightarrow {a^2} = 100 + 100 - 2\left( {100} \right)\left( {\dfrac{{ - 1}}{2}} \right)$,
Now simplifying we get,
$ \Rightarrow {a^2} = 200 - \left( { - 100} \right)$,
Again simplifying we get,
\[ \Rightarrow {a^2} = 200 + 100\],
Now again simplifying we get,
\[ \Rightarrow {a^2} = 300\],
Now taking out the square root we get,
\[ \Rightarrow a = \sqrt {300} \],
Now further simplifying we get,
\[ \Rightarrow a = 10\sqrt 3 \],

So, the length of the side is \[10\sqrt 3 \].

Note: The Cosine Rule is used in the following cases:
1. Given two sides and an included angle (SAS)
2. Given three sides (SSS)
The Cosine Rule states that the square of the length of any side of a triangle equals the sum of the squares of the length of the other sides minus twice their product multiplied by the cosine of their included angle. Some important formulas related to Cosine rule are given below:
${a^2} = {b^2} + {c^2} - 2bc\cos A$,
${b^2} = {a^2} + {c^2} - 2ac\cos B$,
${c^2} = {a^2} + {b^2} - 2ab\cos C$.