Ex.6.4 Q7 Squares and Square Roots - NCERT Maths Class 8
Question
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.
What is known?
Two sides right triangles
What is unknown?
One side of the given triangle.
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
\[\begin{align}{\rm{A}}{{\rm{C}}^{\rm{2}}}\;{\rm{ }}&= \;{\rm{A}}{{\rm{B}}^{\rm{2}}}\;{\rm{ + }}\;{\rm{B}}{{\rm{C}}^{\rm{2}}}\\{{\rm{(13)}}^{\rm{2}}}\;{\rm{ }}&= \;{\rm{A}}{{\rm{B}}^{\rm{2}}}\;{\rm{ + }}\;{{\rm{(5)}}^{\rm{2}}}\\{\rm{169}}\;{\rm{ }}&= \;{\rm{A}}{{\rm{B}}^{\rm{2}}}\;{\rm{ + }}\;{\rm{25}}\\
{\rm{A}}{{\rm{B}}^{\rm{2}}}\;{\rm{ }}&= \;{\rm{169}} - {\rm{25}}\;{\rm{ }}=\;{\rm{144}}\\{\rm{AB}}\;{\rm{ }}&= \;\sqrt {{\rm{144}}} \;{\rm{ }}= \;{\rm{12}}\;{\rm{cm}}\end{align}\]