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D is the mid-point of side BC of DeltaAB...

D is the mid-point of side BC of `Delta`ABC and E is the mid-point of BO. If O is the mid-point of AE, then ar(`Delta`BOE) = 1/k ar(`Delta`ABC). Then k equals

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To solve the problem, we need to determine the value of \( k \) such that the area of triangle \( BOE \) is equal to \( \frac{1}{k} \) times the area of triangle \( ABC \). ### Step-by-Step Solution: 1. **Identify Midpoints**: - Let \( D \) be the midpoint of side \( BC \) in triangle \( ABC \). - Let \( E \) be the midpoint of segment \( BD \). - Let \( O \) be the midpoint of segment \( AE \). 2. **Area of Triangle \( ABD \)**: - Since \( D \) is the midpoint of \( BC \), the median \( AD \) divides triangle \( ABC \) into two equal areas. - Therefore, the area of triangle \( ABD \) is: \[ \text{Area of } \triangle ABD = \frac{1}{2} \text{Area of } \triangle ABC \] 3. **Area of Triangle \( ABE \)**: - Since \( E \) is the midpoint of \( BD \), the median \( AE \) divides triangle \( ABD \) into two equal areas. - Therefore, the area of triangle \( ABE \) is: \[ \text{Area of } \triangle ABE = \frac{1}{2} \text{Area of } \triangle ABD = \frac{1}{2} \left( \frac{1}{2} \text{Area of } \triangle ABC \right) = \frac{1}{4} \text{Area of } \triangle ABC \] 4. **Area of Triangle \( BOE \)**: - Since \( O \) is the midpoint of \( AE \), the median \( BO \) divides triangle \( ABE \) into two equal areas. - Therefore, the area of triangle \( BOE \) is: \[ \text{Area of } \triangle BOE = \frac{1}{2} \text{Area of } \triangle ABE = \frac{1}{2} \left( \frac{1}{4} \text{Area of } \triangle ABC \right) = \frac{1}{8} \text{Area of } \triangle ABC \] 5. **Relate Areas**: - We know from the problem statement that: \[ \text{Area of } \triangle BOE = \frac{1}{k} \text{Area of } \triangle ABC \] - From our calculation, we found that: \[ \text{Area of } \triangle BOE = \frac{1}{8} \text{Area of } \triangle ABC \] - Therefore, we can equate: \[ \frac{1}{k} \text{Area of } \triangle ABC = \frac{1}{8} \text{Area of } \triangle ABC \] 6. **Solve for \( k \)**: - By comparing both sides, we find: \[ \frac{1}{k} = \frac{1}{8} \implies k = 8 \] ### Final Answer: Thus, the value of \( k \) is \( 8 \). ---
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