Question

ABCD 8 is a parallelogram. E and F are the mid-points of the sides AB and AD respectively. Prove that area (ΔAEF) = `1/8` × area (◻ABCD).

Answer 1

Given:

ABCD is a parallelogram.

E is the midpoint of AB.

F is the midpoint of AD.

To Prove:

area (ΔAEF) = `1/8` × area (◻ABCD).

Proof Step-wise:

1. Put A at the origin and use vectors:

Let AB = b and AD = d.

So, vectors AB and AD represent two adjacent sides of the parallelogram.

2. Coordinates of the midpoints:

`E = A + 1/2 AB = b/2`

`F = A + 1/2 AD = d/2`

3. Area of parallelogram ABCD equals the magnitude of the cross product or determinant of b and d:

area (ABCD) = |b × d|

4. Area of triangle AEF equals half the magnitude of the cross product of AE and AF:

`AE = E - A = b/2` 

`AF = F - A = d/2`

`"area" (ΔAEF) = 1/2 xx |AE xx AF|` 

= `1/2 xx |b/2 xx d/2|`

5. Simplify:

`"area" (ΔAEF) = 1/2 × 1/4 × |b × d|`

= `1/8 xx |b xx d|`

6. Substitute area (ABCD) = |b × d|:

area (ΔAEF) = `1/8` × area (ABCD)

Therefore, area (ΔAEF) = `1/8` × area (◻ABCD), as required.