Question

ABCD is a rhombus and ab is produced to E and F such that AB = a b = b prove that BD and ac are perpendicular to each other

Answer 1

Answer:

Proved.

Step-by-step explanation:

Given :

• ABCD is a rhombus .
• AB produced to E and F such that AE = AB = BF.

To Prove :

ED ⊥ FC

Construction :

Join ED and CF and produce it to meet at G.

Proof :

AB is produced to points E and F so, AE = AB = BF ....Eq (1)

Also given,

ABCD is a rhombus .

AB = CD = BC = AD   ....Eq (2)

Now,

In ΔBCF,

BC = BF

__________[ From Equation 1 & 2]

⇒ ∠1 = ∠2

⇒ ∠3 = ∠1 + ∠2  

⇒ ∠3 = 2∠2      ...Eq (3)

Same here,

AE = AD

⇒ ∠5 = ∠6

⇒ ∠4 = ∠5 + ∠6 = 2 ∠5    ....Eq (4)

Now, Addition of Eq 3 & 4.

⇒ ∠4 + ∠3 = 2∠5 + 2∠2

⇒ 180° = 2(∠5 + ∠2)

∴ ∠4 and ∠3 are consecutive interior angles.

⇒ ∠5 + ∠2 = 90°

Therefore, EG ⊥ FC.

Now, in ΔEGF ,

⇒ ∠5 + ∠2 + ∠EGF = 180°

⇒ 90° + ∠EGF = 180°      

⇒ ∠EGF = 180° - 90°

⇒ ∠EGF = 90°

_______[ Hence proved. ]