Question

1. ABC is an isosceles triangle with AB = AC and BD, CE are its two medians.
Show that BD = CE​

Answer 1

For this we have to first make the two triangles congruent
So we will triangles ABD and ACE
So the conditions will be-
1. angleA (common)
2.BE=CE(given)
3.angle E=angle D
So, AAS
Thus,ABD is congruent to ACE
So,BD=CE(Cpct)


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Answer 2

Question :

ABC is an isosceles ∆ in which AB = AC and BD , CE are it's two medians . Show that BD = CE .

ANSWER

Given :

ABC is an isosceles ∆ in which AB = AC and BD , CE are it's two medians .

Required to prove :

• BD = CE

Congruency axiom used :

• SAS ( Side , Angle , Side )

Proof :

ABC is an isosceles ∆ in which AB = AC and BD , CE are it's two medians .

We need to prove that ;

• BD = CE

So,

Let's consider the ∆ ABC .

In ∆ ABC ,

AB = AC are equal .

so,

∠B = ∠C

[ Reason : If corresponding sides are equal corresponding angles are also equal ]

BE = AE

AD = DC

[ Reason : The altitude divides the side into 2 equal halves ]

However,

Consider ∆ ADB & ∆ ACE

In ∆ ADB & ∆ ACE

This implies ;

➳ AB = AC

[ Reason : Since, ∆ ABC is an isosceles triangle ]

➳ ∠BAD = ∠CAE

[ Reason : Common angle between the 2 triangles ]

➳ AD = AC

[ Reason : Since, BD is a median ]

By , SAS Congruency rule

∆ ADB $\cong$ ∆ ACE

Hence,

BD = CE

[ Reason : Corresponding Parts of congruent triangles ]

Alternate method :

Consider ∆ BEC & ∆ BDE .

In ∆ BEC & ∆ BDE

➳ BC = BC

[ Reason : Common side between the 2 triangles ]

➳ ∠EBC = ∠DCB

[ Reason : since, ∆ ABC is an isosceles triangle ]

Since,

AB = BC

BE = ½ AB , DC = ½ BC

½ AB = ½ BC

➳ BE = DC

By, SAS Congruency rule

∆ BEC $\cong$ ∆ BDE

Hence,

BD = CE

[ Reason : Corresponding Parts of congruent triangles ]

Hence Proved !

Additional Information :

Question :

If two triangles are similar then do they have same measurements ?

Answer :

No, when two triangles are said to be similar it doesn't mean that they have same measurements . But the ratio of the corresponding sides is constant/equal .

Question :

What are all the congruency axioms ?

Answer :

The congruency axioms are ;

SSS ( side , side , side )

SAS ( side , angle , side )

AAS ( Angle , Angle , Side )

SSA ( side , side , Angle )

ASA ( Angle , Angle , Angle )

AAA ( Angle , Angle , Angle )

Question :

What is the difference between SAS and SSA .

Answer

The basic difference between SAS and SSA is ;

SAS - Side , Angle included between the two sides , Side

SSA - Side , Angle , Side

In SAS , the angle should be included between the two sides which are taken under the congruency part .

Whereas,

In SSA , the angle need not to be an included angle .

The same logic is also for ASA & AAS .