Ex.6.4 Q7 Squares and Square Roots - NCERT Maths Class 8

In a right triangle \(\rm{}ABC\), \(\rm{}∠B = 90°\).

(i) If \(AB = 6 \,\rm{ cm}\)\(\), \(BC = 8\, \rm{cm}\)\(\), find \(AC\) 

(ii) If \(AC = 13\,\rm{cm},\) \(\) \(BC = 5 \,\rm{cm} \)\(\), find \(AB \)

Text Solution

What is known?

Two sides right triangle

What is unknown?

One side of the given triangle

Reasoning:

In a right-angle triangle if two sides are given then third side can be calculated using the Pythagoras theorem.

Steps:

(i)

\(AB = 6\;\rm{cm}\) \(BC = 8\;\rm{cm} \)  \(AC=\) ?

According to Pythagoras theorem

\[\begin{align}{\rm{A}}{{\rm{C}}^{\rm{2}}}\,{\rm{ }}&= \;{\rm{A}}{{\rm{B}}^{\rm{2}}}\;{\rm{ + }}\;{\rm{B}}{{\rm{C}}^{\rm{2}}}\\{\rm{A}}{{\rm{C}}^{\rm{2}}}\;{\rm{ }}&= \;{{\rm{(6)}}^{\rm{2}}}\;{\rm{ + }}\;{{\rm{(8)}}^{\rm{2}}}\\{\rm{A}}{{\rm{C}}^{\rm{2}}}\;{\rm{ }}&= \;{\rm{100}}\\\;\,{\rm{AC}}\;{\rm{ }}&= \;\sqrt {{\rm{100}}} \\\;\,{\rm{AC}}\;{\rm{ }}&= \;{\rm{10}}\;{\rm{cm}}\end{align}\]

(ii) 

\(AC = 13\;\rm{cm} \) , \(BC = 5\;\rm{cm}\) , \(AB \)=?

According to Pythagoras theorem

\[\begin{align}{\rm{A}}{{\rm{C}}^{\rm{2}}}\;{\rm{ }}&= \;{\rm{A}}{{\rm{B}}^{\rm{2}}}\;{\rm{ + }}\;{\rm{B}}{{\rm{C}}^{\rm{2}}}\\{{\rm{(13)}}^{\rm{2}}}\;{\rm{ }}&= \;{\rm{A}}{{\rm{B}}^{\rm{2}}}\;{\rm{ + }}\;{{\rm{(5)}}^{\rm{2}}}\\{\rm{169}}\;{\rm{ }}&= \;{\rm{A}}{{\rm{B}}^{\rm{2}}}\;{\rm{ + }}\;{\rm{25}}\\
{\rm{A}}{{\rm{B}}^{\rm{2}}}\;{\rm{ }}&= \;{\rm{169}} - {\rm{25}}\;{\rm{ }}=\;{\rm{144}}\\{\rm{AB}}\;{\rm{ }}&= \;\sqrt {{\rm{144}}} \;{\rm{ }}= \;{\rm{12}}\;{\rm{cm}}\end{align}\]