question_answer 1)

Which of the following can be the sides of a right triangle? (i) \[\text{2}.\text{5 cm}\],              \[\text{6}.\text{5 cm}\],           \[\text{6 cm}\]. (ii) \[\text{2 cm}\],                \[\text{2 cm}\],            \[\text{5 cm}\]. (iii) \[\text{1}.\text{5 cm}\],              \[\text{2 cm}\],            \[\text{2}.\text{5 cm}\]. In the case of right-angled triangles, identify the right angles.

Answer:

                (i)            \[\mathbf{2}\mathbf{.5 cm, 6}\mathbf{.5 cm, 6 cm}\] We see that \[\text{(2}.\text{5)2}+{{\text{6}}^{\text{2}}}=\text{6}.\text{25}+\text{36}=\text{42}.\text{25}={{(\text{6}.\text{5})}^{\text{2}}}\]Therefore, the given lengths can be the sides of a right triangle. Also, the angle between the lengths, 2.5 cm and 6 cm is a right angle. (ii)           \[\mathbf{2 cm, 2 cm, 5 cm}\] \[\because \]     \[\text{2}+\text{2}=\text{4/}>\text{5}\] \[\therefore \]  The given lengths cannot be the sides of a triangle                 \[\left| \begin{align}   & \text{The sum of the lengths of any two sides of a} \\  & \text{triangle is greater than the third side} \\ \end{align} \right.\] (iii)          \[\mathbf{1}\mathbf{.5 cm, 2 cm, 2}\mathbf{.5 cm}\] We find that \[\text{1}.{{\text{5}}^{\text{2}}}+{{\text{2}}^{\text{2}}}=\text{ 2}.\text{25}+\text{4}=\text{6}.\text{25}=\text{ 2}.{{\text{5}}^{\text{2}}}\] Therefore, the given lengths can be the sides of a right triangle. Also, the angle between the lengths 1.5 cm and 2 cm is a right angle.