Question

Point M is the point of intersection of all the 3 medians of a triangle\(\triangle\)ABC. The median drawn from vertex A intersects the side BC at point D and the lengths of AD and BC are 21 cm and 24 cm respectively. What is the sum of the lengths of BD and MD?

• A. 21 cm
• B. 22 cm
• C. 20 cm
• D. 19 cm

Answer 1

The Correct Option is D

Centroid and Segment Calculation 

Step 1: Understanding the Problem

Point M is the centroid of the triangle, which divides each median in a 2:1 ratio, with the longer segment closer to the vertex.

Step 2: Given Information

• Length of median AD = 21 cm
• Length of side BC = 24 cm

Step 3: Calculate Segment AM (Vertex to Centroid)

Using the centroid property:

\[ AM = \frac{2}{3} \times AD = \frac{2}{3} \times 21 = 14 \text{ cm} \]

Step 4: Calculate Segment MD (Centroid to D)

Using the centroid property:

\[ MD = \frac{1}{3} \times AD = \frac{1}{3} \times 21 = 7 \text{ cm} \]

Step 5: Calculate Segment BD

Since \( BD + CD = BC \) and \( BD = BC - CD \), we have:

\[ BD = 24 - 7 = 17 \text{ cm} \]

Step 6: Calculate the Sum of BD and MD

The total length is:

\[ BD + MD = 17 + 7 = 19 \text{ cm} \]

Step 7: Conclusion

The sum of the lengths of BD and MD is 19 cm.