In a right angle triangle ABC, $\angle B = {90^\circ }$, find
$\left( i \right)$ If AB = 6 cm, BC = 8 cm, find AC
$\left( {ii} \right)$ If AC = 13 cm, BC = 5 cm, find AB.

- Hint: In this particular question first draw the pictorial representation of the given problem it will give us a clear picture of what we have to find out then later in the solution use the property of Pythagoras theorem so use these concepts to reach the solution of the question.

Complete step-by-step answer:

Given data:
ABC is a right triangle with angle B = 90 degrees.
Consider the right angle triangle ABC, with $\angle B = {90^\circ }$ as shown in the figure.
In the above diagram, Hypotenuse = AC, Perpendicular = AB and base = BC.
So by Pythagoras theorem we have,
$ \Rightarrow {\left( {{\text{Hypotenuse}}} \right)^2} = {\left( {{\text{perpendicular}}} \right)^2} + {\left( {{\text{base}}} \right)^2}$
Now substitute the variables we have,
$ \Rightarrow {\left( {{\text{AC}}} \right)^2} = {\left( {{\text{AB}}} \right)^2} + {\left( {{\text{BC}}} \right)^2}$........................... (1)
$\left( i \right)$ If AB = 6 cm, BC = 8 cm, find AC
Now from equation (1) we have,
$ \Rightarrow {\left( {{\text{AC}}} \right)^2} = {\left( {{\text{AB}}} \right)^2} + {\left( {{\text{BC}}} \right)^2}$
Now substitute all the values in the above equation we have,
$ \Rightarrow {\left( {{\text{AC}}} \right)^2} = {\left( {\text{6}} \right)^2} + {\left( {\text{8}} \right)^2}$
Now simplify it we have,
$ \Rightarrow {\left( {{\text{AC}}} \right)^2} = 36 + 64 = 100 = {10^2}$
So the length of the hypotenuse, AC = 10 cm.
$\left( {ii} \right)$ If AC = 13 cm, BC = 5 cm, find AB.
Now from equation (1) we have,
$ \Rightarrow {\left( {{\text{AC}}} \right)^2} = {\left( {{\text{AB}}} \right)^2} + {\left( {{\text{BC}}} \right)^2}$
Now substitute all the values in the above equation we have,
$ \Rightarrow {\left( {{\text{13}}} \right)^2} = {\left( {{\text{AB}}} \right)^2} + {\left( {\text{5}} \right)^2}$
$ \Rightarrow {\left( {{\text{AB}}} \right)^2} = {\left( {13} \right)^2} - {\left( {\text{5}} \right)^2}$
Now simplify it we have,
$ \Rightarrow {\left( {{\text{AB}}} \right)^2} = 169 - 25 = 144 = {\left( {12} \right)^2}$
So the length of the perpendicular, AB = 12 cm.
So this is the required answer.

Note: Whenever we face such types of questions the key concept we have to remember is that according to the Pythagoras theorem square of the hypotenuse is equal to the sum of the square of the two legs of the right angle triangle, so just substitute the values in the Pythagoras equation and simplify as above we will get the required answer.