Question

In $△XYZ,\angle Y={90}^{\circ }$, if the lengths of the medians drawn from $X$ and $Z$ are $5$ and $\sqrt{40}$, find the length of the hypotenuse

• A. $10$
• B. $\sqrt{13}$
• C. $2\sqrt{40}$
• D. $2\sqrt{13}$

Answer 1

The correct option is D $2\sqrt{13}$

Given, $\angle Y={90}^{\circ }$

$XQ=5$ and $ZP=\sqrt{40}$

Let $XY=x$ and $YZ=y$

$\therefore XP=PY=\frac{x}{2}$ and $YQ=QZ=\frac{y}{2}$

$Z{X}^2=x^2+y^2$

In $△ZPY$

$Z{P}^2=P{Y}^2+Y{Z}^2$

$(\sqrt{40}{)}^2=(\frac{x}{2}{)}^2+(y{)}^2$

$⟹x^2+4y^2=160$ -------(i)

In $△XYQ$

$X{Q}^2=X{Y}^2+Y{Q}^2$

$(5{)}^2=(x{)}^2+(\frac{y}{2}{)}^2$

$⟹4x^2+y^2=100$ -------(i)

Solving (i) and (ii) we get

$x=4$ and $y=6$

$Hypotenuse=XZ=\sqrt{x^2+y^2}=\sqrt{4^2+6^2}$

$=2\sqrt{13}$

Option D.