Question

If the midpoints of the sides of a kite are joined, prove that the formed quadrileteral is a rectangle​

Answer 1

Step-by-step explanation:

Let us consider the kite

A

B

C

D

as shown below. In this we have midpoints

P

,

Q

,

R

,

S

as midpoints of sides

A

B

,

A

D

,

B

C

and

C

D

respectively. Le us also join

B

D

and

A

C

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Now in isosceles triangle

Δ

A

B

D

as

P

is midpoint of

A

B

and

Q

is midpoint of

A

D

,

P

Q

=

1

2

B

D

and

P

Q

||

B

D

.

Smilarly in

Δ

B

C

D

,

R

S

=

1

2

B

D

and

R

S

||

B

D

,

and hence,

P

Q

||

R

S

and

P

Q

=

R

S

Similarly we can prove that

P

R

=

Q

S

and

P

R

||

Q

S

Now using

S

S

S

postulate, we can prove

Δ

A

B

C

≡

Δ

A

C

D

and hence

∠

B

A

C

=

∠

D

A

C

.

Therefore

A

C

is bisector of isosceles triangle

Δ

A

B

D

and hence

A

C

⊥

B

D

and

P

Q

⊥

P

R

Hence quadrilateral

P

Q

R

S

is a rectangle.