Question

The vertex A of triangle abc is joined tio point d on side bc the mid point of ad is e prove that ar(triangle bec) = 1/2 ar(triangle abc)

Answer 1

Since, E is the mid point of AD, so CE is the median for ΔADC and BE is the median for Δ ABD.

Since a median divides a triangle in two congruent parts,

ar (ΔAEC) = ar (ΔCDE) (1)

ar (ΔAEB) = ar (ΔBED) (2)

Adding (1) and (2), we get

ar (ΔAEC) +ar (ΔAEB) = ar (ΔCDE) + ar (ΔBED)

ar (ΔABC) – ar (ΔBEC) = ar (ΔBEC)

ar (ΔABC) = 2ar (ΔBEC)

⇒ar (ΔBEC) = ar (ΔABC)

Hence proved.

Answer 2

Hello Mate!

Given : E is mid point on AD.

To prove : ar(∆BEC) = ½ ar(∆ABC)

Proof : Since AE = DE ( E was mid point )

Hence, BE is median in ∆ABD.

Since median divides triangles into two equal areas therefore,

ar(∆ABE) = ar(∆BED) __(i)

Again, since AE = DE ( E was mid point )

Hence, CE is median in ∆ACD.

Since median divides triangles into two equal areas therefore,

ar(∆AEC) = ar(∆CED) __(ii)

On adding (i) and (ii) we get,

ar(∆ABE) + ar(∆AEC) = ar(∆BED) + ar(∆CED)

ar(quad ABEC) = ar(∆BEC)

ar(quad ABEC) + ar(∆BEC) = ar(∆ABC)

ar(∆BEC) + ar(∆BEC) = ar(∆ABC)

2ar(∆BEC) = ar(∆ABC)

ar(∆BEC) = ½ ar(∆ABC)

ʜᴇɴᴄᴇ ᴘʀᴏᴠᴇᴅ

Have great future ahead!