question_answer

The medians of a right triangle which are drawn from the vertices of the acute angles are \[\text{5 cm}\] and \[\text{2}\sqrt{10}\text{ cm}\]. The hypotenuse of the triangle is:

A) \[\text{10 cm}\]

B) \[\text{2}\sqrt{40}\text{ cm}\]

C) \[\sqrt{13}\text{ cm}\]

D) \[2\sqrt{13}\text{ cm}\]

Correct Answer: D

Solution :

[d]

In right angled\[\Delta ABC,\]

\[A{{D}^{2}}+C{{E}^{2}}=A{{B}^{2}}+B{{D}^{2}}+B{{C}^{2}}+B{{E}^{2}}\]

\[=A{{B}^{2}}+\frac{1}{4}B{{C}^{2}}+B{{C}^{2}}+\frac{1}{4}A{{B}^{2}}\]

\[\Rightarrow \,\,\,\,\,\,4(A{{D}^{2}}+C{{E}^{2}})=5(A{{B}^{2}}+B{{C}^{2}})\]

\[\Rightarrow \,\,\,\,\,\,4(25+40)=5{{(AC)}^{2}}\]

\[\Rightarrow \,\,\,\,\,\,A{{C}^{2}}=\frac{4\times 65}{5}\]

\[\Rightarrow \,\,\,\,\,\,AC=2\sqrt{13}cm.\]