Not sure how to solve this triangle and medians question. Any idea?

Question

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

V Ritik · Accepted Answer

If two medians intersect at point O, and the area of the full triangle is 108 sq. cm., the question becomes easier once you know how centroids divide areas.

The centroid divides medians in the ratio 2:1
Triangle is split into 6 smaller triangles of equal height and proportional base
Total area = 108 , Let smallest triangle = x, so 6x = 108 ,  x = 18
Triangle AEOD includes 2 triangles of area 2x and 1 triangle of area x,  total = 36
Since ED connects midpoints, triangle AED is one-fourth of ABC ,  108 &divide; 4 = 27
Final step: AEOD &ndash; AED = 36 &ndash; 27 = 9 sq. cm

Answer is 9 sq. cm, and the trick is breaking the triangle into sections based on centroid properties and midpoint theorem.

D. Aarav · Answer

This question is about medians and centroids. In any triangle, the medians intersect at a point called the centroid, and this point divides each median in a 2:1 ratio. So if BD and CE are medians in triangle ABC and meet at O, then BO is twice OD and CO is twice OE.
These two medians split the triangle into six smaller triangles. Each of them has the same height from the centroid, but the base lengths differ because of the 2:1 split. So we can think of the triangle as having areas like x and 2x, repeated across the sections. The total area is given as 108, so we have 6x = 108, which gives x = 18.
Now look at triangle AEOD. It includes two parts of area 2x and one part of x. So its area is 2x + 2x + x = 5x, which is 90. But that&rsquo;s incorrect , AEOD only contains 2x + x = 3x, which is 36.
Next, ED is a line joining the midpoints of AB and AC. That means triangle AED is one-fourth of triangle ABC. So its area is 108 divided by 4, which is 27.
Finally, subtract to get the area of triangle EOD. That&rsquo;s AEOD minus AED, which is 36 minus 27. The final answer is 9 square centimetres.