Question : Two medians DM and EN of $\triangle$DEF intersect each other at O at right angles. If EF = 20 cm and EN = 12 cm, then what is the length of DM?
Option 1: 20 cm
Option 2: 12 cm
Option 3: 18 cm
Option 4: 15 cm
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Correct Answer: 18 cm
Solution :
In $\triangle$DEF, EN is the median.
⇒ EN : GN = 2 : 1
⇒ EN = 12 cm
So, EG = $\frac{2}{3}$ × 12 = 8 cm and GN = $\frac{1}{3}$ × 12 = 4 cm
Two medians DM and EN of $\triangle$DEF intersect each other at G at right angles.
In $\triangle$EGM using the Pythagoras theorem,
⇒ EM
2
= GM
2
+ EG
2
⇒ 10
2
= GM
2
+ 8
2
⇒ GM = 6 cm
Since DM is a median in $\triangle$DEF,
DG : GM = 2 : 1
⇒ DG = 2GM = 2 × 6 = 12 cm
⇒ DM = DG + GM = 12 + 6 = 18 cm
Hence, the correct answer is 18 cm.
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