Question

In the adjoining figure, ABCD and ABFE are two parallelograms.

Prove that: area (ABCD) + area (ABFE) = area (EFCD).

Answer 1

Given: ABCD and ABFE are parallelograms as shown.

To Prove: area (ABCD) + area (ABFE) = area (EFCD).

Proof [Step-wise]:

1. Choose coordinates (vector set-up).

Put A at the origin O.

Let AB = u.   ...(a nonzero vector along the horizontal)

Let AD = p and AE = r.

Then points are:

B = A + u

D = A + p

E = A + r

C = B + p = A + u + p

F = B + r = A + u + r

2. Identify vectors for sides of parallelogram EFCD.

EF = F – E

= (A + u + r) – (A + r)

= u

FC = C – F

= (A + u + p) – (A + u + r)

= p – r

So, EFCD is a parallelogram with adjacent side vectors u and (p – r).

3. Express areas by the magnitude of the 2D cross product or base × height.

area (ABCD) = |u × p|   ...(Parallelogram with adjacent sides u and p)

area (ABFE) = |u × r|   ...(Parallelogram with adjacent sides u and r)

area (EFCD) = |u × (p – r)| = |u × p – u × r|   ...(By bilinearity of the cross product).

4. Use the geometry of the picture:

ABCD and ABFE lie on opposite sides of AB.

So, the oriented scalar values u × p and u × r have opposite signs. 

Therefore, u × p – u × r = (u × p) + |u × r| and taking absolute value gives |u × (p – r)| = |u × p| + |u × r|.

5. Substitute the area expressions from step 3: 

area (EFCD) = |u × (p – r)|

= |u × p| + |u × r|

= area (ABCD) + area (ABFE).