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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