In triangle ABC, medians AD and BE intersect at point G, the centroid. If D is the midpoint of BC and E is the midpoint of AC, then what is the ratio of the area of triangle BDG to the area of quadrilateral GDCE?

Introduction / Context:
This problem explores how medians divide a triangle into smaller regions with specific area relationships. In triangle geometry, medians are segments joining a vertex to the midpoint of the opposite side, and their point of intersection is the centroid. The centroid has important properties, including dividing each median in a 2 : 1 ratio and partitioning the triangle into regions of equal or related areas. This question asks you to find the ratio between the area of one small triangle and the area of a larger quadrilateral formed by the medians inside triangle ABC.

Given Data / Assumptions:
• Triangle ABC is a non degenerate triangle.
• D is the midpoint of BC, so AD is a median.
• E is the midpoint of AC, so BE is a median.
• The medians AD and BE intersect at point G, the centroid of triangle ABC.
• We are asked to find the ratio of area(ΔBDG) to area(quadrilateral GDCE).

Concept / Approach:
A key concept is that the three medians of a triangle divide it into six smaller triangles of equal area. Although only two medians are explicitly mentioned, we can rely on this general fact to reason about the areas. Triangle BDG is one of those small triangles. Quadrilateral GDCE can be seen as the union of two of those small triangles, which leads naturally to a simple area ratio. Using symmetry and the equal area property rather than numeric coordinates keeps the solution conceptual and elegant, but a coordinate geometry approach would also confirm the same ratio.

Step-by-Step Solution:
Step 1: Recall that the three medians of a triangle intersect at the centroid and divide the triangle into 6 smaller triangles of equal area. Step 2: In triangle ABC, medians AD, BE and the third median from C (even if not drawn in the statement) partition the triangle into 6 congruent area regions. Step 3: Triangle BDG is one of those 6 small regions, so area(ΔBDG) is equal to 1/6 of the area of triangle ABC. Step 4: The quadrilateral GDCE can be viewed as the union of two of those small triangles, each having area equal to area(ΔBDG). Step 5: Therefore, quadrilateral GDCE has area equal to 2 * area(ΔBDG). Step 6: Hence the ratio area(ΔBDG) : area(GDCE) = 1 : 2.

Verification / Alternative check:
As an alternative, place triangle ABC on a coordinate plane with simple coordinates for A, B and C, find the midpoints D and E, determine the centroid G, and then compute the areas using the coordinate area formula. Such a calculation confirms that triangle BDG has area equal to one third of triangle BDC and that GDCE is composed of two equal small triangles each having the same area as BDG. Numerical verification matches the conceptual reasoning and again yields the ratio 1 : 2, so the result is stable regardless of the specific shape of triangle ABC.

Why Other Options Are Wrong:
Option 1 : 3 would imply that the quadrilateral has three times the area of triangle BDG, which would require it to contain three of the equal area small triangles, but it actually contains only two. Option 2 : 3 and 3 : 4 are ratios that do not match the equal six part structure produced by the medians, and a careful breakdown of the interior regions shows they are not consistent with the way areas are divided. Only 1 : 2 accurately reflects that GDCE is composed of exactly two regions whose areas are equal to that of BDG.

Common Pitfalls:
A common mistake is to assume without justification that the medians divide the triangle only into four regions or that these regions do not have equal area. Some learners also misidentify which smaller regions make up the quadrilateral GDCE and mistakenly count three or more small triangles instead of two. Remembering the theorem that three medians form 6 equal area triangles is crucial, and carefully visualising or sketching the triangle with all medians drawn helps avoid miscounting. Breaking down complex shapes into known smaller equal parts is a powerful geometry strategy.

Final Answer:
The required ratio of areas is 1 : 2.