IOQM 2023 Question 5 centroid triangle

IOQM 2023 Q5: In a triangle ABC , let E be the midpoint of AC and F be the midpoint of AB .
The medians BE and CF intersect at G . Let Y and Z be the midpoints of BE and CF respectively. If the area of triangle ABC is 480 , find the area of triangle GYZ .

Answer 10: 

Solution:
1. 3 medians intersecting at centroid divide the triangle into 6 triangles of equal area.
2. GBC has 2 of those triangles so its area is 1/3 of the total area.
3. Look at GYZ. GB is 2/3 of the median BE. BY is 1/2 of the median BE. So GY = 2/3 - 1/2 = 1/6BE. GYZ and GBC are similar with side ratio of 1:4. So [GYZ] = 1/16[GBC] = 1/16 * 1/3 [ABC] = 480/16*3 = 10.


Prerequisites:
Centroid and median properties in triangle.
Similarity criterion for triangles.

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