Ex.6.5 Q1 Triangles Solution - NCERT Maths Class 10

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Question

Sides of triangles are given below. Determine which of them are right triangles.

In case of a right triangle, write the length of its hypotenuse.

(i)  \(7 \;\rm{cm}, 24 \;\rm{cm}, 25\; \rm{cm}\) (ii)  \(3\; \rm{cm}, 8 \;\rm{cm}, 6 \;\rm{cm}\) (iii)  \(50\; \rm{cm}, 80 \;\rm{cm}, 100 \;\rm{cm}\) (iv)  \(13\; \rm{cm}, 12 \;\rm{cm}, 5 \;\rm{cm}\)

Text Solution

Reasoning:

As we know,in a triangle, the square of one side is equal to the sum of the squares of the other two sides,then the angle opposite the first side is a right angle.

Steps:

(i) \({{(25)}^{2}} =625\)

\[\begin{align} {{7}^{2}}+{{24}^{2}}&=49+576 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&=625 \\ \therefore \,\,\,\,{{(25)}^{2}}&={{7}^{2}}+{{24}^{2}} \\ \end{align}\]

Length of hypotenuse = \(25\;\rm{cm}\)

(ii) ) \({{8}^{2}} =64\)

\[\begin{align} {{3}^{2}}+{{6}^{2}}&=9+36 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&=45 \\ \,\,\,\,\,\,\,\,\,\,\,\,{{8}^{2}}&\ne {{3}^{2}}+{{6}^{2}} \\ \end{align}\]

(iii)\({{(100)}^{2}}=1000\)

\[\begin{align} {{50}^{2}}+{{80}^{2}}&=2500+6400 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&=8900 \\ \,\,\,\,\,\,\,{{(100)}^{2}}&\ne {{50}^{2}}+{{80}^{2}} \\ \end{align}\]

(iv)\({{\left( 13 \right)}^{2}}=169\)

\[\begin{align} {{12}^{2}}+{{5}^{2}}&=144+25 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&=169 \\ \end{align}\]

\(\therefore{{(13)}^{2}}={{12}^{2}}+{{5}^{2}}\)

Length of hypotenuse = \(13\;\rm{cm}\)

\(\Rightarrow \) (i) and (iv) are right triangle.