LET ABC BE THE TRIANGLE . AD IS THE MEDIAN. IN TRIANGLE ABD AND ACD AD =AD (COMMON SIDE) ANGLE ABD =ACD (EACH IS 90)AB =AC .THUS TRIANGLE ABD CONGRUENT TO ACD BY SAS CONGRUENCE. THIS IMPLIES MEDIAN DIVIDES A TRIANGLE INTO TWO EQUAL PARTS

- -6

Consider a triangle *ABC* Let *D* be the midpoint of , *E* be the midpoint of , *F* be the midpoint of , and *O* be the centroid.

By definition, . Thus[*A**D**O*] = [*B**D**O*],[*A**F**O*] = [*C**F**O*],[*B**E**O*] = [*C**E**O*], and , where [*A**B**C*] represents the area of triangle ; these hold because in each case the two triangles have bases of equal length and share a common altitude from the (extended) base, and a triangle's area equals one-half its base times its height.

We have:

Thus, and

Since , therefore, . Using the same method, you can show that

- -2

ABC is the triangle and AD be the median (which means BD =DC).

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = ½ AN x BD [1]

ar(ACD) = ½ AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

=> Median of a triangle divides it into two triangles of equal area.

- 112

ABC is the triangle and AD be the median (which means BD =DC).

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

=> Median of a triangle divides it into two triangles of equal area

- 24

ABC is the triangle and AD be the median (which means BD =DC).

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

=> Median of a triangle divides it into two triangles of equal areaABC is the triangle and AD be the median (which means BD =DC).

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

=> Median of a triangle divides it into two triangles of equal areaABC is the triangle and AD be the median (which means BD =DC).

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

=> Median of a triangle divides it into two triangles of equal areaABC is the triangle and AD be the median (which means BD =DC).

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = � AN x BD [1]

ar(ACD) = � AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

=> Median of a triangle divides it into two triangles of equal area

- -27

Construct AE perpendicular to BC.

To prove- Ar (ABD)=Ar (ACD)

Proof- Ar (ABD)=1/2*base*height

=1/2*AE*BD-(1)

Similarly,Ar (ACD)=1/2*AE*CD-(2)

According to given data-BD=CD

hence from(1)and(2)

Ar (ABD)=Ar (ACD)

So it is proved that median divides it into two triangles of equal area

- -1

**ABC is the triangle and AD be the median (which means BD =DC).**

**Construct AN perpendicular (90 degree) to BC.**

**Now ar(ABD) = ½ AN x BD [1]**

**ar(ACD) = ½ AN x CD [2]**

**We know that BD=CD ( since AD is the median)
Therefore ,from [1] and [2]
ar ( ABD) = ar(ACD)**

**=> Median of a triangle divides it into two triangles of equal area.**

- -5

- 3

Construct AN perpendicular (90 degree) to BC.

Now ar(ABD) = ? AN x BD [1]

ar(ACD) = ? AN x CD [2]

We know that BD=CD ( since AD is the median)

Therefore ,from [1] and [2]

ar ( ABD) = ar(ACD)

=> Median of a triangle divides it into two triangles of equal area.

- 0