Question : In $\triangle X Y Z$, L and M are the middle points of the sides XY and XZ, respectively. N is a point on the segment LM such that LN : NM = 1 : 2. If LN = 5 cm, then YZ is equal to:
Option 1: 30 cm
Option 2: 24 cm
Option 3: 28 cm
Option 4: 26 cm
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Correct Answer: 30 cm
Solution :
In ΔXYZ, L and M are the middle points of the sides XY and XZ, respectively.
LN : NM = 1 : 2 and
LN = 5 cm
In $\triangle$XYZ and in $\triangle$XLM
$\angle$X = $\angle$X [common]
$\angle$XYZ = $\angle$XLM
$\angle$XZY = $\angle$XML
$\therefore \triangle$XYZ ~ $\triangle$XLM
LN : NM = 1 : 2
LN = 5 cm
$\frac{\text{LN}}{\text{NM}} = \frac{1}{2}$
$⇒\frac{5}{\text{NM}} = \frac{1}{2}$
$⇒\text{NM} = 10$ cm
LM = LN + NM = 5 + 10 = 15 cm
Since, $\triangle$XYZ ~ $\triangle$XLM
$\frac{\text{LM}}{\text{YZ}}=\frac{\text{XL}}{\text{XY}}$
$⇒\frac{15}{\text{YZ}}=\frac{\text{XL}}{\text{2XL}}$ [∵ L is midpoint of XY]
$⇒\frac{15}{\text{YZ}}=\frac{1}{2}$
$\therefore\text{YZ} = 30$ cm
Hence, the correct answer is 30 cm.
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