How do you solve the right angle ABC given a=2, c=7?

Hint: Draw a right angle triangle ABC with ‘a’ as base, ‘b’ as perpendicular and ‘c’ as hypotenuse. Then apply Pythagoras' theorem for the triangle ABC i.e. ${{c}^{2}}={{a}^{2}}+{{b}^{2}}$. Put the values of ‘a’ and ‘c’ from the given data and do the necessary calculations to obtain the value of ‘b’.

Complete step-by-step solution:

ABC is a right angle triangle with sides a, b and c.
Pythagoras' theorem: In a right angle triangle the square of the hypotenuse is equal to the summation of the square of the base and perpendicular.
For the above triangle, ‘a’ is the base, ‘b’ is the perpendicular and ‘c’ is the hypotenuse. So, the Pythagoras' theorem can be applied as ${{c}^{2}}={{a}^{2}}+{{b}^{2}}$
Now, we have a=2 and c=7
Using Pythagoras' theorem and putting the values of ‘a’ and ‘c’, we get
$\begin{align}
  & {{c}^{2}}={{a}^{2}}+{{b}^{2}} \\
 & \Rightarrow {{\left( 7 \right)}^{2}}={{\left( 2 \right)}^{2}}+{{b}^{2}} \\
 & \Rightarrow 49=4+{{b}^{2}} \\
 & \Rightarrow {{b}^{2}}=49-4 \\
 & \Rightarrow b=\sqrt{45} \\
 & \Rightarrow b=3\sqrt{5} \\
\end{align}$
This is the required solution of the given question.

Note: The value of ‘b’ can be simplified as $\sqrt{45}=\sqrt{9\times 5}=\sqrt{9}\times \sqrt{5}=3\sqrt{5}$. a, b and c should be taken as base, perpendicular and hypotenuse respectively. Altering these values will affect the value of ‘b’ in the final result. Using Pythagoras' theorem and doing the necessary calculations the value of ‘b’ should be obtained.