Suppose the medians BD and CE of a triangle ABC intersect at a point O. If area of triangle ABC is 108 sq. cm., then, the area of the triangle EOD, in sq. cm., is

Question

Accepted Answer

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1743576697/aptidude_questions/asuhtpsu7oddfnmb5efr.png)

1. **Centroid Properties and Division of Area:**  
   - The centroid divides each median in the ratio of 2:1.  
   - It also divides the entire triangle into six smaller triangles of equal area.

2. **Area Division in Triangle ABC:**  
   - The total area of triangle ABC = 108 sq. cm.  
   - Each of the six smaller triangles formed by the medians has an area:  
     $$
     	ext{Area of each smaller triangle} = \frac{108}{6} = 18 	ext{ sq. cm.}
     $$

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1743576703/aptidude_questions/pssnje6l4pgihxxllgla.png)

3. **Area of Triangle AEOD (Trapezium AEDC):**  
   - Since BD is a median, it divides triangle ABC into two equal areas:  
     $$
     	ext{Area of } 	riangle ABD = 	ext{Area of } 	riangle BDC = \frac{108}{2} = 54 	ext{ sq. cm.}
     $$  
   - Similarly, CE divides triangle ABC into two equal areas:  
     $$
     	ext{Area of } 	riangle AEC = 	ext{Area of } 	riangle EBC = 54 	ext{ sq. cm.}
     $$

4. **Area of Quadrilateral AEDC (Midpoint Connection):**  
   - The line joining midpoints divides the triangle into two smaller triangles such that:  
     $$
     	ext{Area of } 	riangle AED = \frac{	ext{Area of } 	riangle ABC}{4} = \frac{108}{4} = 27 	ext{ sq. cm.}
     $$

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1743576709/aptidude_questions/z19todzqzal9mltemv0w.png)

5. **Calculating Area of Triangle EOD:**  
   - From the total area of AEOD (36 sq. cm.), subtract the area of AED (27 sq. cm.):  
     $$
     	ext{Area of } 	riangle EOD = 36 - 27 = 9 	ext{ sq. cm.}
     $$

Thus, the area of triangle **EOD is 9 sq. cm**.