Question

In triangle LMN, LM = 8cm, MN = 6cm and LMN = 90°
X and Y are the midpoints of MN and LN respectively.
Find YXN and YN.​

Answer 1

$\textbf{Converse of Basic proportionality theorem:}$

$\text{If a line divides two sides of a triangle in the same ratio,}$

$\text{then the line is parallel to the third side}$

$\textbf{Given:}$

$\text{In \triangle LMN, LM=8cm, MN=6 cm \angle{LMN}=90^{\circ} }$

$\textbf{To find:}$

$\angle{YXN}\;\text{and}\;YN$

$\text{In right angled \triangle LMN, by pythagoras theorem}$

$LN^2=LM^2+MN^2$

$LN^2=8^2+6^2$

$LN^2=64+36$

$LN^2=100$

$LN=10\;\text{cm}$

$\text{Since X and Y are midpoints of MN and LN,}$

$MX=XN=3\;\text{cm}\;\text{and}\;NY=LY=5\;\text{cm}$

$\implies\dfrac{NX}{MX}=1\;\text{and}\;\dfrac{NY}{LY}=1$

$\implies\bf\dfrac{NX}{MX}=\dfrac{NY}{LY}$

$\text{By converse of basic proportionality theorem, }$

$\text{XY is parallel to ML}$

$\implies\bf\angle{YXN}=90^{\circ}$ $\text{( \because corresponding angles are equal})$

$\therefore\textbf{YN=5 cm and \bf\angle{YXN}=90^{\circ} }$

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