Question

8. If a line joining the vertices A and
AABD and ABCD intersects the base!
point O, prove that:
AO
area of AABD
area of ABCD
CO
Ar
B
С
[Hint: Draw perpendiculars from A and
diagonal BD.​

Answer 1

Answer:

Step-by-step explanation:

⇒  In given figure ABCD is a square having all sides are equal and opposite sides parallel to each other. AC and BD are diagonals.

⇒  In △ABC and △BAD,

⇒  AB = AB                             [Common line]

⇒  BC = AD                             [Sides of square are equal.]

⇒  ∠ABC = ∠BAD                  [All four angles of square is 90  

∘

]

⇒  △ABC ≅ △BAD              [By SAS property]

⇒  In a △ OAD and △OCB,

⇒  AD = CB                      [sides of a square]

⇒  ∠OAD = ∠OCB    [Alternate angle]

⇒  ∠ODA = ∠OBC    [Alternate angle]

⇒  △OAD ≅ △OCB  [By ASA Property]

⇒  So, OA = OC                   -----  ( 1 )

⇒  Similarly, OB = OD         ------ ( 2 )

From ( 1 ) and ( 2 ) we get that AC and BD bisect each other.

⇒  Now, in △OBA and △ODA,

⇒  OB = OD                       [From ( 2 )]

⇒  BA = DA

⇒  OA = OA                       [Common line]

⇒  ∠AOB + ∠AOD       ----- ( 3 )        [By CPCT]

⇒  ∠AOB + ∠AOD = 180  

∘

      [Linear pair]

⇒  2∠AOB = 180  

∘

 

∴   ∠AOB = ∠AOD = 90  

∘

 

∴  We have proved that diagonals of square are equal and perpendicular to each other.