The document provides two examples of geometric proofs involving areas of shapes. The first example proves that a triangle's median divides it into two equal areas. It does so by drawing an altitude and showing the two resulting triangles have the same base and altitude. The second example proves the area of a quadrilateral ABCD equals the area of triangle ADE by showing two triangles on the same base and between parallel lines have equal areas.

1. PRESENTATION TO SHOW

2. EXAMPLE- 1

3. Show that a median of
a triangle divides it into
two triangles of equal
areas.

5. Let ABC be a triangle and let AD be
one of its medians
suppose, you wish to show
area(ABD) = area(ACD)
 Since the formula for area involves altitude ,let us draw
AN┴BC.
Now area(ABD) = 1 ∕ 2 x base x altitude(of ∆ADB)
= 1 ∕ 2 x BD x AN
= 1 ∕ 2 x CD x AN(As BD = CD)
= 1 ∕ 2 x base x altitude(of ∆ACD)
= area(ACD)

6. EXAMPLE- 2

7. ABCD is a quadrilateral and
BE||AC and also BE meets
DC produced at E. Show that
area of ∆ADE is equal to the
area of the quadrilateral
ABCD.

9. Solution: In the given figure ∆BAC and ∆EAC lie on the same base
AC and between the same parallels AC and BE.
Therefore, area(BAC) = area(EAC)(two triangles on the
same base and between the same parallels are equal in area)
So, area(BAC) +area(ADC) = area(EAC) + area(ADC)
(adding same areas on both sides)
area(ABCD) = area(ADE)