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Home / Geometry / ABC is an isosceles triangle and AB = AC. The side BC is extended up to D. Prove that AD>AB.

ABC is an isosceles triangle and AB = AC. The side BC is extended up to D. Prove that AD>AB.

Solution: General enunciation: ABC is an isosceles triangle and AB = AC. The side BC is extended up to D. Prove that AD>AB.

tr1

Particular enunciation: Given that, ABC is an isosceles triangle and AB = AC. The side BC is extended up to D. It is required to prove that AD>AB.

Proof: In ABC AB = AC

∴∠ABC = ∠ACB

Now ∠ACD be an exterior angle of ABC

∴∠ACD >interior opposite ∠ABC

⟹∠ACD >∠ACB       [∵∠ABC = ∠ACB]

Again ∠ACB is the external angle of ACD.

∴∠ACB>∠ADC

Since ∠ACD >∠ACB and ∠ACB>∠ADC

∴∠ACD >∠ADC

Now In ACD

∠ACD >∠ADC

⟹AD>AC

⟹AD>AB                            [AB = AC]

∴ AD>AB  (Proved)

Problem: In the ABC, AB = AC and D is any point on BC. Prove that AB>AD.

Solution: General equation: In the ABC, AB = AC and D is any point on BC. Prove that AB>AD.

tr2

Particular Enunciation: Given that, in the ABC, AB = AC and D is any point on BC. It is required to prove that AB>AD.

Proof: Given that in ABC, AB = AC

∴∠ABC = ∠ACB   [∵ the angles opposite to equal sides are equal]

Again, ∠ADB is an exterior angle of ACD

∴∠ADB>interior opposite angle ∠ACD

⟹∠ADB>∠ACB

⟹∠ADB>∠ABC    [∵∠ABC = ∠ACB]

⟹∠ADB>∠ABD

Now in triangle ABD,

∠ADB>∠ABD             [∵ the angles opposite to equal sides are equal]

⟹AB>AD  (Proved)

Problem: In the ABC, AB ⊥ AC and D is any point on AC. Prove that BC>BD.

Solution: General enunciation: In the ABC, AB ⊥ AC and D is any point on AC. Prove that BC>BD.

tr3

Particular enunciation: Given that, in the ABC, AB ⊥ AC and D is any point on AC. It is required to prove that BC>BD.

Proof: Given that, in ABC, AB ⊥ AC

∠A = 900

∴ BC be the hypotenuse.

∴ ∠BAC > ∠BAC

Again ∠BDC is an exterior angle of ABD.

∴∠BDC>interior opposite ∠BAD

⟹∠BDC>∠BAC

⟹∠BDC>∠BCA                   [∵∠BAC > ∠BAC]

⟹∠BDC>∠BCD

Now, in BCD

∠BDC>∠BCD

⟹BC>BD               [∵ the angles opposite to equal sides are equal]

∴ BC>BD (Proved)

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