Question : In $\triangle ABC$, D and E are the midpoints of sides BC and AC, respectively. AD and BE intersect at G at the right angle. If AD = 18 cm and BE=12 cm, then the length of DC (in cm ) is:
Option 1: 10
Option 2: 6
Option 3: 9
Option 4: 8
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Correct Answer: 10
Solution : Given: D is the midpoint of BC E is the midpoint of AC AD and BE intersect at G at right angles. AD = 18 cm and BE = 12 cm Concept used: The centroid of a triangle divides the median in the ratio of 2 : 1. Calculation: According to the question, AG : GD = 2 : 1 and BG : GE = 2 : 1 ⇒ AG = $(\frac{2}{3})$ × AD = $(\frac{2}{3})$ × 18 = 12 cm ⇒ GD = $(\frac{1}{3})$ × AD = $(\frac{1}{3})$ × 18 = 6 cm Similarly, ⇒ BG = $(\frac{2}{3})$ × BE = $(\frac{2}{3})$ × 12 = 8 cm ⇒ GF = $(\frac{1}{3})$ × BE = $(\frac{1}{3})$ × 12 = 4 cm Now in ΔBGD, $\angle$ BGD = 90° By Pythagoras theorem, BD$^2$ = BG$^2$ + GD$^2$ ⇒ BD$^2$ = 8$^2$ + 6$^2$ ⇒ BD$^2$ = 10$^2$ ⇒ BD = 10 cm We know that, BD = DC = 10 cm [D is the midpoint of BC] Hence, the correct answer is 10 cm.
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