Question

ABCD is a rectangle. AC and BD are the diagonals of the rectangle, which cross at point E. a) prove that both diagonals divide the rectangle into two congruent triangles

Answer 1

Answer :-

Given :-

ABCD is a rectangle

AC and BD are the diagonals of the rectangle which intersect at point E .

Required to prove :-

• Diagonals divide the rectangle into two congruent triangles

Proof :-

It is given that :-

ABCD is a rectangle .

In which ,

AC and BD are the diagonals which intersect at point " E " .

It is needed to prove that diagonals divide the rectangle into 2 congruent triangles .

So,

ABCD is a rectangle

AC is a diagonal

Consider ∆ ABC and ∆ ADC

In ∆ ABC and ∆ ADC

AC = AC [ Reason :- common side ]

AB = CD [ Reason :- In a rectangle opposite sides are equal ]

BC = AD [ Reason :- In a rectangle opposite sides are equal ]

Hence,

By using S.S.S. axiom

∆ ABC ≅ ∆ ADC

So,

Diagonal AC divides the rectangle into 2 congruent triangles

Similarly,

In rectangle ABCD

BD is the diagonal

So,

Consider ∆ ABD and ∆ BCD

In ∆ ABD and ∆ BCD

BD = BD [ Reason : Common side ]

AB = CD [ Reason : opposite sides are equal in a rectangle ]

AD = BC [ Reason : opposite sides are equal in a rectangle ]

Hence ,

By using S.S.S. axiom

∆ ABD ≅ ∆ BCD

So,

Diagonal BD also divides the rectangle into 2 congruent triangles .

Therefore,

The both diagonals of the rectangle divide the rectangle into 2 congruent triangles .

Hence Proved

Points to remember :-

1. properties of a rectangle :-

• Opposite sides are equal

• All angles are equal to 90°

• Diagonals bisect each other

2. We can solve this problem using any other congruency axioms except A.A.A. axiom

3. Note :- If two triangles are said to be congruent then their areas are also equal .