Question

show that the bisectors of a parallelogram forms a rectangle.

please solve completely not that similarly wali line....​

Answer 1

Answer:

LMNO is a parallelogram in which bisectors of the angles L, M, N, and O intersect at P, Q, R and S to form the quadrilateral PQRS. Hence the angle bisectors of a parallelogram form a rectangle as all the angles are right angles; we conclude that it IS RECTANGLE. Hence proved.

Answer 2

$\huge\fbox \pink{A}\fbox \purple{n}\fbox \blue{s}\fbox \pink{w}\fbox \purple{e}\fbox \blue{r}$

Data : ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively.

To prove : PQRS is a rectangle.

Construction : Diagonals AC and BD are drawn.

Proof : To prove PQRS is a rectnagle, one of its angle should be right angle.

In ∆ADC, S and R are the mid points of AD and DC.

∴ SR || AC

SR = 1212AC (mid-point formula)

In ∆ABC, P and Q are the mid points AB and BC.

∴ PQ || AC PQ = ½AC.

g ∴ SR || PQ and SR = PQ

$\small{\boxed{\sf{{∴ PQRS is a parallelogram. }}}}$

But diagonals of a rhombus bisect at right angles. 90° angle is formed at ‘O’.

∴ ∠P = 90°

∴ PQRS is a parallelogram, each of its angle is right angle.

This is the property of rectangle.

$\small{\boxed{\sf{{∴ PQRS is a rectangle.}}}}$

Hᴏᴘᴇ Tʜɪs Hᴇʟᴘs Yᴏᴜ

AP bisects ∠A and DP bisects ∠D.

Therefore ,

∠DAP + ∠ADP = ½ ∠A + ½ ∠D

= ½ (∠A + ∠D)

⇒ ∠DAP + ∠ADP = ½ x 180° = 90° (sum of consecutive ∠s of a parallelogram)

$\small{\boxed{\sf{{ ∠DAP + ∠ADP = 90° ….① }}}}$

Also in ∆ADP,

∠DAP + ∠ADP + ∠APD = 180° (angle sum property of a triangle)

⇒ 90° + ∠APD = 180° ⇒ ∠APD = 90° [using (ii)] But ∠QPS = ∠APD (vertically opposite angles)

⇒ $\small{\boxed{\sf{{∠QPS = 90° ...② }}}}$

Similarly, BR bisects ∠B and CR bisects,

∠C, therefore ,

$\small{\boxed{\sf{{∠QRS = 90° …③}}}}$

Also, AQ bisects ∠A and BQ bisects ∠B and DS bisects ∠D and CS bisects ∠C,

Therefore ,

$\small{\boxed{\sf{{ ∠PQR = 90° …④ }}}}$

$\small{\boxed{\sf{{ And ∠PSR = 90° ...⑤ }}}}$

So, using ②,③,④ and ⑤

we can say that PQRS is a quadrilateral in which all angles i.e. ∠P, ∠Q, ∠R and ∠S are right angles.

$\LARGE{\boxed{\sf{{Hence, PQRS is a rectangle.}}}}$

Hᴏᴘᴇ Tʜɪs Hᴇʟᴘs Yᴏᴜ