How do you use the Pythagorean Theorem to find the missing side of the right triangle with the given measures: $b = 5,c = 8$?

Hint: In the question above, we have a right-angle triangle, and we have two of its side measurements, we are supposed to find out one of the sides with the help of Pythagoras theorem. We all are aware of the Pythagoras theorem, which involves the hypotenuse value when a triangle has a right angle. With the help of this theorem, we can easily find the value of the two sides.

Formula used: ${(side1)^2} + {(side2)^2} = {(Hypotenuse)^2}$

Complete step-by-step solution:
We have a right-angle triangle with sides $b = 5,c = 8$, and we are supposed to find one of its sides.
We all are aware that,
\[ \Rightarrow {a^2} + {b^2} = {c^2}\]
Right, so that is the equation of Pythagorean Theorem. So, let plug in the numbers of b and c.
\[ \Rightarrow {a^2} + {5^2} = {8^2}\]
So, we have our values of $b$ and $c$.
Well, let's simplify things further by squaring the numbers we have already.
\[ \Rightarrow {a^2}\; + {\text{ }}25{\text{ }} = {\text{ }}64\]
We will now subtract $25$ from both side which gives us:
$ \Rightarrow {a^2} = 39$
Now we can square root the answer.
$ \Rightarrow \sqrt {{a^2}} = \sqrt {39} $
Now, we are left with this:
$ \Rightarrow a = \sqrt {39} $
So, the final answer is $\sqrt {39} $ or the decimal value of $\sqrt {39} $ is \[6.244997998\].

Therefore, for a right-angle triangle with sides $b = 5,c = 8$ the last side will be of the measurement $\sqrt {39} $ is \[6.244997998\].

Note: A hypotenuse side is the longest side of a triangle. A hypotenuse basically is in front of the right angle of the triangle, and is always to the opposite side of the angle with the measurement of ${90^ \circ }$. A Pythagoras theorem involves the two sides and the hypotenuse of the right-angled triangle, and tells that the addition of the squares of the sides is equal to the square of the hypotenuse.