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in Parallelograms by (44.3k points)

The vertex A of △ ABC is joined to a point D on the side BC. The midpoint of AD is E. Prove that ar (△ BEC) = ½ ar (△ ABC).

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Best answer

From the figure we know that BE is the median of △ ABD

So we get

Area of △ BDE = Area of △ ABE

It can be written as

Area of △ BDE = ½ (Area of △ ABD) …… (1)

From the figure we know that CE is the median of △ ADC

So we get

Area of △ CDE = Area of △ ACE

It can be written as

Area of △ CDE = ½ (Area of △ ACD) …….. (2)

By adding both the equations

Area of △ BDE + Area of △ CDE = ½ (Area of △ ABD) + ½ (Area of △ ACD)

By taking ½ as common

Area of △ BEC = ½ (Area of △ ABD + Area of △ ACD)

So we get

Area of △ BEC = ½ (Area of △ ABC)

Therefore, it is proved that ar (△ BEC) = ½ ar (△ ABC).

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