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

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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

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