Question

In the following figure, AB =BC and AD = CD. Show that BD bisects AC at right angles.​

Answer 1

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In △ABD and △CBD,

AD=CD [ given ]

BD=BD [ Common ]

AB=BC [ given ]

Hence, △ABD≅△CBD [ SSS ]

∠ABD=∠1

∠CBD=∠2

∠1=∠2 [ CPCT ]

Also ,

∠3=∠4 [ CPCT ]

Hence ,

BD bisects both the angles ABC and ADE.

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Answer 2

Question-

In the following figure, AB = BC and AD = CD. Show that BD bisects AC at right angles.

Answer-

We are required to prove ∠BEA = ∠BEC = 90° and AE = EC.Consider ∆ABD and ∆CBD,

AB = BC (Given)

AD = CD (Given)

BD = BD (Common)

Therefore, ∆ABD ≅ ∆CBD (By SSS congruency)

∠ABD = ∠CBD (CPCT)

Now, consider ∆ABE and ∆CBE,

AB = BC (Given)

∠ABD = ∠CBD (Proved above)

BE = BE  (Common)

Therefore, ∆ABE≅ ∆CBE (By SAS congruency)

∠BEA = ∠BEC (CPCT)

And ∠BEA +∠BEC = 180° (Linear pair)

2∠BEA = 180° (∠BEA = ∠BEC)

∠BEA = 180°/2 = 90° = ∠BEC

AE = EC (CPCT)

Hence, BD is a perpendicular bisector of AC.