Question -
Answer -
Given:
(1) ABC is a triangle
(2) AD is the median of ╬ФABC
(3) G is the midpoint of the median AD
To prove:
(a) Area of ╬Ф ADB = Area of ╬Ф ADC
(b) Area of ╬Ф BGC = 2 Area of ╬Ф AGC
Construction: Draw a line AM perpendicular to AC
Proof: Since AD is the median of ╬ФABC.
Therefore BD = DC
So multiplying by AM on both sides we get
In ╬ФBGC, GD is the median
Since the median divides a triangle in to two triangles ofequal area. So
Area of┬а├ДBDG┬а= Area of┬а├ДGCD
тЗТ┬аAreaof┬а├ДBGC┬а=2(Area of┬а├ДBGD)
Similarly In ╬ФACD, CG is the median
тЗТ┬аAreaof┬а├ДAGC┬а=Area of┬а├ДGCD
From the above calculation we have
Area of┬а├ДBGD┬а= Area of┬а├ДAGC
But Area of┬а├ДBGC┬а= 2(Area of┬а├ДBGD)
So we have
Area of┬а├ДBGC┬а= 2(Area of┬а├ДAGC)
Hence it is proved that
(1)┬а
(2)┬а