Question : Two medians JX and KY of $\triangle \mathrm{JKL}$ intersect each other at Z at right angles. If KL = 22 cm and KY = 12 cm, then what is the length of JX?
Option 1: $6 \sqrt{19} \mathrm{~cm}$
Option 2: $3 \sqrt{57} \mathrm{~cm}$
Option 3: $2 \sqrt{57} \mathrm{~cm}$
Option 4: $4 \sqrt{19} \mathrm{~cm}$
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Correct Answer: $3 \sqrt{57} \mathrm{~cm}$
Solution :
Two medians JX and KY of $\triangle$JKL intersect each other at Z at right angles.
KL = 22 cm and KY = 12 cm
The medians of a triangle divide each other in a 2:1 ratio.
According to the question,
Since JX is median, KX = XL = $\frac{22}{2}$ = 11 cm
Since JX and KY intersected each other at 90º, $\angle$KZX = 90º.
Hence, $\triangle$KZX is a right-angled triangle.
According to the concept,
XZ = $\sqrt{11^2-8^2}$
XZ = $\sqrt{57}$
Since, JZ : XZ = 2 : 1, JX = $\frac{3}{1} \times \sqrt{57}$ = $3 \sqrt{57}$ cm
Hence, the correct answer is $3 \sqrt{57} \mathrm{~cm}$.
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