Question

2. In a triangle ABC, E is the mid-point of median
AD. Show that ar (BED)= - ar(ABC).​

Answer 1

Answer:

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Step-by-step explanation:

AD is the median of ΔABC. Therefore, it will divide ΔABC into two triangles of equal area.

∴ Area (ΔABD) = Area (ΔACD)

⇒Area (ΔABD ) = (½) area (Δ ABC) ------------(1)

In ΔABD, E is the mid-point of AD.

Therefore, BE is the median.

∴ Area (ΔBED) = Area (ΔABE)

Area (ΔBED) = (1/2)Area (ΔABD)

Area (ΔBED) = (½ ) x(1/2) Area (ΔABC) [From (1)]

∴ Area (ΔBED) = (1/4)Area (ΔABC).

Answer 2

Solution:

ar(BED) = (1/2)×BD×DE

Since, E is the mid-point of AD,

AE = DE

Since, AD is the median on side BC of triangle ABC,

BD = DC

DE = (1/2) AD — (i)

BD = (1/2)BC — (ii)

From (i) and (ii), we get,

ar(BED) = (1/2)×(1/2)BC × (1/2)AD

⇒ ar(BED) = (1/2)×(1/2)ar(ABC)

⇒ ar(BED) = ¼ ar(ABC)