In a triangle ABC, E is the mid-point of median AD. Show that ar (BED) = 1/4 ar(ABC).

Solution:

We know that the median of a triangle divides it into two triangles of equal areas. AD is a median for triangle ABC and BE is the median of ΔABD. 

Since AD is the median of ΔABC, so it will divide ΔABC into two equal triangles.

∴ ar (ΔABD) = ar (ΔADC)

Also, ar (ΔABD) = 1/2 ar(ABC)   .....(i)

Now, In ΔABD, BE is the median,

Therefore, BE will divide ΔABD into two equal triangles

ar (ΔBED) = ar (ΔBAE) and ar (ΔBED) = 1/2 ar(ΔABD)

ar (ΔBED) = 1/2 × [1/2 ar(ABC)] (Using equation (i))

∴ ar (ΔBED) = 1/4 ar(ΔABC)

☛ Check: NCERT Solutions Class 9 Maths Chapter 9

Video Solution:

In a triangle ABC, E is the mid-point of median AD. Show that ar (BED) = 1/4 ar(ABC)

Maths NCERT Solutions Class 9 Chapter 9 Exercise 9.3 Question 2

Summary:

If E is the mid-point of the median AD of triangle ΔABC, then Area of (ΔBED) = 1/4 Area of (ΔABC).

☛ Related Questions:

• Show that the diagonals of a parallelogram divide it into four triangles of equal area.
• In Fig. 9.24, ABC and ABD are two triangles on the same base AB. If line- segment CD is bisected by AB at O, show that ar(ABC) = ar (ABD).
• D, E and F are respectively the mid-points of the sides BC, CA and AB of a ΔABC. Show that: i) BDEF is a parallelogram. ii) ar (DEF) = 1/4 ar (ABC) iii) ar (BDEF) = 1/2 ar (ABC)
• In Fig. 9.25, diagonals AC and BD of quadrilateral ABCD intersect at O such that OB = OD. If AB = CD, then show that:i) ar (DOC) = ar (AOB)ii) ar (DCB) = ar (ACB)iii) DA || CB or ABCD is a parallelogram.[Hint: From D and B, draw perpendiculars to AC.]