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

Answer 1

Step 1: Area Ratio of Triangles △ABD and △BDC

Given that BD is a median, it divides △ABC into two triangles of equal area. Thus, \[ \text{Area of } △ABD = \text{Area of } △BDC \] Therefore, \[ \text{Area of } △ABD = \frac{1}{2} \text{Area of } △ABC \] Assuming the area of △ABC is \( 108 \, \text{sq cm} \), we have: \[ \text{Area of } △ABD = 54 \, \text{sq cm} \]

Step 2: Area Ratio of Triangles △EDB and △ADE

Similarly, since CE is a median, it divides △ABC into two triangles of equal area. Point E is the midpoint of AC. Thus, AE = EC. Therefore, the median DE in triangle △ADC divides it into two triangles of equal area. However, we are given the ratio of areas of △EDB and △ADE. Considering △ABD, if DE is a segment from D to the midpoint E of AC, it doesn't necessarily imply that △EDB and △ADE have equal areas without further information about E in relation to AB. The provided information implies that E is the midpoint of AC, and D is the midpoint of BC. Thus CE and BD are medians. The statement "Given: \[ \text{Area of } △EDB : \text{Area of } △ADE = 1 : 1 \]" implies E is the midpoint of AC, and DE is a segment within △ABD. Assuming E is the midpoint of AC and D is the midpoint of BC, then △ADE and △CDE have equal areas. Also △BDE and △CDE have equal areas. Therefore △ADE = △BDE. Given △ABD = △EDB + △ADE = 54: \[ \text{Area of } △ADE = \frac{54}{2} = 27 \, \text{sq cm} \]

Step 3: Using the Centroid Property

The centroid \( O \) is the intersection of the medians. The centroid divides each median in a 2:1 ratio. In △ADC, EO is part of the median CE. In △ABD, DO is part of the median BD. The problem states that O is the intersection of BD and CE. O lies on BD, and O lies on CE. The statement "In triangle △ADE, point O lies on median" is incorrect as O is on median BD and median CE. The centroid property states that the medians divide the triangle into six smaller triangles of equal area. Alternatively, the centroid divides the triangle into three larger triangles of equal area (△AOB, △BOC, △COA). However, the provided step uses a different property. If O is the centroid, then DO:OB = 1:2. In △ADE, O is a point. The relationship between O and △ADE needs clarification based on the centroid property of △ABC. Assuming the statement refers to the property that medians divide the triangle area in halves, and the centroid divides the median in 2:1, consider △ADE. If O is the centroid, it lies on median BD. The statement implies that O divides a median within △ADE, which is not directly true unless O is a vertex of △ADE. However, if we consider △AEC, and BO is a line segment, the centroid property relates areas around the centroid. The statement \[ \text{Area of } △BEO : \text{Area of } △EOD = 2 : 1 \] implies that O divides the segment BE in a 2:1 ratio which is incorrect for centroid O and median BE. Let's reinterpret Step 3 using the centroid property correctly. The centroid divides each median in a 2:1 ratio. Thus, BO:OD = 2:1 and CO:OE = 2:1. Consider △ABD. DO is 1/3 of median BD. This implies Area(△ADO) = Area(△ABO) = Area(△BDO) = 1/3 Area(△ABD). This is also incorrect. The property is that the centroid divides the triangle into 6 triangles of equal area. Let's assume the statement \[ \text{Area of } △BEO : \text{Area of } △EOD = 2 : 1 \] is derived from the centroid property applied to median CE in triangle △ABC. This means O divides CE such that CO:OE = 2:1. This implies that the area of △COB = 2 * Area(△EOB), and Area(△COA) = 2 * Area(△EOA). The given relation \( \text{Area of } △BEO : \text{Area of } △EOD = 2 : 1 \) implies that O lies on BE and divides it in a 2:1 ratio, which is not standard. However, if we assume that the areas of △BOD and △EOD are related due to the centroid, and the ratio 2:1 is correct, then: Let Area(△EOD) = \( x \). Then Area(△BEO) = \( 2x \). We are given △ABD = 54, and △ABD = △ADE + △EDB. If △ADE = △EDB, then △ADE = 27. If O is the centroid, then Area(△AOD) = Area(△BOD) = Area(△COD) = 1/3 Area(△ABC) = 108/3 = 36. This contradicts the given information. Let's restart assuming the provided ratios are derived from a specific geometric configuration or property not fully explained. Given △ABD = 54. Given Area(△EDB) : Area(△ADE) = 1 : 1. This means △ABD = △EDB + △ADE = 2 * △ADE = 54. So, △ADE = 27. Given Area(△BEO) : Area(△EOD) = 2 : 1. This implies that O is a point that divides the segment BD such that the ratio of areas of triangles with vertex E and bases BO and OD is 2:1. This is unusual. Let's assume the intended property is related to the centroid dividing medians. If O is the centroid, then DO:OB = 1:2. Consider △ADE. If we take OD as a base, and E as the vertex, then Area(△EOD) and Area(△EOB) should be related if O lies on BD. The relation Area(△BEO) : Area(△EOD) = 2 : 1 can arise if O divides BD in a 2:1 ratio from B, i.e., BO:OD = 2:1. This means O is the centroid. If O is the centroid, then OD = 1/3 BD. Consider △ABE. E is the midpoint of AC. So BE is a median of △ABC. O is on BE. If O is the centroid, then O divides median BE such that BO:OE = 2:1. This implies Area(△BOD) = 2 * Area(△EOD). Let's go back to the initial steps, which seem consistent. △ABD = 54. △ADE = 27. The statement "In triangle △ADE, point O lies on median" is likely a misunderstanding. O is the centroid of △ABC. Consider △ACE. O is on median CE. Consider △ABD. O is on median BD. The statement "Area of △BEO : Area of △EOD = 2 : 1" is critical. If O is the centroid, then BO:OD = 2:1. This means OD = 1/3 BD. In △ADE, we have Area(△ADE) = 27. Let's consider △ABD. OD is a segment from D to O on the median BD. This is not making sense. Let's assume the statement "Area of △BEO : Area of △EOD = 2 : 1" is correct and O is a point on BD. Let Area(△EOD) = \( x \). Then Area(△BEO) = \( 2x \). This means Area(△BDE) = Area(△BEO) + Area(△EOD) = \( 2x + x = 3x \). We previously found Area(△ADE) = 27 and Area(△EDB) = 27. So, △BDE = △EDB = 27. Therefore, \( 3x = 27 \), which gives \( x = 9 \). Area(△EOD) = \( x = 9 \, \text{sq cm} \). This aligns with the final answer. The assumption is that O lies on BD and the ratio of areas △BEO and △EOD is 2:1. This implies that BO:OD = 2:1 if E was the vertex, which is consistent with O being the centroid. Final Answer: \[ \boxed{9 \, \text{sq cm}} \]