Question

if ad is medium of a triangle ABC then prove that angle ADB and triangle ADC are equal in area if g is the midpoint of median ad prove that ar(∆bgc)=2ar(∆agc)

Answer 1

Hello Mate!

Given : AD is median in ∆ABC and G is mid point on AD.

To prove : 1. ar(∆ADB) = ar(∆ADC)
2. ar(∆BGC) = 2ar(∆AGC)

Proof : Since both the triangles, ∆ADB and ∆ADC lie on equal bases, BD = CD ( as AD is median ) and will have same altitude so,

ar(∆ADB) = ar(∆ADC)

Now, in ∆BGC also GD is median

ar(∆BGD) = ar(∆CGD) __(i)

In ∆ABD, BG is median

ar(∆AGB) = ar(∆BGD) __(ii)

Similarly in ∆ACD, CG is median

ar(∆AGC) = ar(∆CGD) __(iii)

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

ar(∆AGC) = ar(∆AGB)

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

ar(∆AGB) + ar(∆AGC) = ar(∆BGD) + ar(∆CGD)

ar(∆AGC) + ar(∆AGC) = ar(∆BGC)

2ar(∆AGC) = ar(∆BGC)

ʜᴇɴᴄᴇ ᴘʀᴏᴠᴇᴅ

Have great future ahead!

Answer 2

$\huge \bold \red{dear \: friend}$
here is your answer OK ☺☺☺☺☺☺☺☺



median divides triangle into two equal parts of equal area

so, area of triangle adb=area of triangle adc..

if g is the mid point of ad
then gd is median of triangle cgb
so area of cgd= area of dgb........(i)

in triangle adc, cg is the median
so, area of triangle agc=area of triangle cdg...... (ii)

from equation (i) and (ii)
area of dgb= area agc (iii)

adding area of triangle agc in equation (iii)

so, area of triangle agc + area of triangle agc= area of triangle dgb + area of triangle agc


2area agc= area of dgb+ area of cdg

as area of triangle agc= area of triangle cdg

2area agc = area of triangle bgc

hence proved.....