In triangle $$ riangle PQR$$, two medians $$PS$$ and $$RT$$ intersect at point $$G$$, forming a right angle. If the lengths of $$PS$$ and $$RT$$ are 9 cm and 6 cm respectively, calculate the length of segment $$RS$$.

Question

In triangle $$	riangle PQR$$, two medians $$PS$$ and $$RT$$ intersect at point $$G$$, forming a right angle. If the lengths of $$PS$$ and $$RT$$ are 9 cm and 6 cm respectively, calculate the length of segment $$RS$$.

Accepted Answer

To find the length of segment $$RS$$, we first calculate the segments divided by the medians at point $$G$$. Since $$G$$ is the centroid, it divides the medians in a ratio of $$2:1$$.

For median $$PS$$:  
$$GS = \frac{1}{3} 	imes PS = \frac{1}{3} 	imes 9 = 3 \, 	ext{cm}.$$

For median $$RT$$:  
$$RG = \frac{2}{3} 	imes RT = \frac{2}{3} 	imes 6 = 4 \, 	ext{cm}.$$

Now, using the Pythagorean theorem in right triangle $$	riangle RGS$$:  
$$RS = \sqrt{GS^2 + RG^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, 	ext{cm}.$$

Thus, the length of $$RS$$ is **5 cm**.

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