Question : Two medians NA and OB of $\triangle \mathrm{NOP}$ intersect each other at S at right angles. If NA = 15 cm and OB = 15 cm, then what is the length of OA?
Option 1: $5 \sqrt{5} $ cm
Option 2: $7 \sqrt{5}$ cm
Option 3: $6 \sqrt{5}$ cm
Option 4: $3 \sqrt{5}$ cm
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Correct Answer: $5 \sqrt{5} $ cm
Solution :
Given,
Median NA = 15 cm
Median OB = 15 cm
Medians intersect at right angles at point S
Theorem: The medians of a triangle intersect each other at the centroid, which divides each median in the ratio 2 : 1.
As S is the point where the medians intersect, it divides each median in a 2 : 1 ratio, with the longer section towards the midpoint of the side.
$\therefore$ the length of OS (longer section of OB) = $\frac{2}{3} ×$15 = 10 cm
And, SA (shorter section of NA) = $\frac{1}{3} ×$15 = 5 cm.
Since the medians intersect at right angles
Applying the Pythagorean theorem,
⇒ OA
2
= OS
2
+ SA
2
⇒ OA
2
= (10 cm)
2
+ (5 cm)
2
⇒ OA
2
= 100 cm
2
+ 25 cm
2
⇒ OA
2
= 125 cm
2
⇒ OA = $\sqrt{125}$ cm
2
⇒ OA = 5$\sqrt5$ cm
Hence, the correct answer is $5\sqrt5$ cm.
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