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Delta ABC and Delta DBC are equal in a...

`Delta ABC and Delta DBC ` are equal in area and the points `A` and `D` lie on the same side of `BC` . Then

A

AD is perpendicular to AB

B

AD is parpendicular to DC

C

AD is parallel to BC

D

AD is parallel to the line passing through the point of intersection of AC and BD

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The correct Answer is:
To solve the problem, we will analyze the given information about triangles ABC and DBC, which are equal in area and have points A and D on the same side of line BC. ### Step-by-Step Solution: 1. **Understand the Given Information**: We have two triangles, ΔABC and ΔDBC, which are equal in area. Points A and D are located on the same side of line BC. 2. **Draw the Triangles**: Start by sketching triangle ABC with vertices A, B, and C. Then, sketch triangle DBC with vertices D, B, and C. Note that BC is the common base for both triangles. 3. **Identify the Base and Heights**: In both triangles, the base BC is the same. Let the height from point A to line BC be H1, and the height from point D to line BC be H2. 4. **Write the Area Formulas**: The area of triangle ABC can be expressed as: \[ \text{Area}_{ABC} = \frac{1}{2} \times BC \times H1 \] The area of triangle DBC can be expressed as: \[ \text{Area}_{DBC} = \frac{1}{2} \times BC \times H2 \] 5. **Set the Areas Equal**: Since the areas of the two triangles are equal, we can set the area formulas equal to each other: \[ \frac{1}{2} \times BC \times H1 = \frac{1}{2} \times BC \times H2 \] 6. **Simplify the Equation**: We can cancel out \(\frac{1}{2}\) and \(BC\) (assuming \(BC \neq 0\)) from both sides: \[ H1 = H2 \] 7. **Interpret the Result**: The equality \(H1 = H2\) means that the perpendicular distances from points A and D to line BC are equal. This indicates that both points A and D are at the same distance from line BC. 8. **Conclude About the Lines**: Since A and D are at the same distance from line BC and lie on the same side of it, the line segment AD must be parallel to line BC. Thus, we conclude that line AD is parallel to line BC. ### Final Answer: The correct conclusion is that line AD is parallel to line BC.
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