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
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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