Show that if the diagnals of a quadrilateral intersect each other at r

1.	Show that if the diagnals of a quadrilateral intersect each other at right angld , then the sum of squares of opposite sides are equal
Answer» ANSWER: Sum of SQUARES of opposite sides are equal Step-by-step explanation: Given that ABCD is a square. To prove : AC=BD and AC and BD bisect each other at right angles. Proof: (i) In a ΔABC and ΔBAD, AB=AB ( COMMON line) BC=AD ( opppsite sides of a square) ∠ABC=∠BAD ( = 90° ) ΔABC≅ΔBAD( By SAS property) AC=BD ( by CPCT). (ii) In a ΔOAD and ΔOCB, AD=CB ( opposite sides of a square) ∠OAD=∠OCB ( transversal AC ) ∠ODA=∠OBC ( transversal BD ) ΔOAD≅ΔOCB (ASA property) OA=OC ---------(i) Similarly OB=OD ----------(ii) From (i) and (ii) AC and BD bisect each other. Now in a ΔOBA and ΔODA, OB=OD ( from (ii) ) BA=DA OA=OA ( common line ) ΔAOB=ΔAOD----(iii) ( by CPCT ∠AOB+∠AOD=180° (linear pair) 2∠AOB=180° ∠AOB=∠AOD=90° ∴AC and BD bisect each other at right angles. please mark me as BRAINLEST

Show that if the diagnals of a quadrilateral intersect each other at right angld , then the sum of squares of opposite sides are equal

Answer»

Step-by-step explanation:

Given that ABCD is a square.

To prove : AC=BD and AC and BD bisect each other at right angles.

Proof:

(i) In a ΔABC and ΔBAD,

AB=AB ( COMMON line)

BC=AD ( opppsite sides of a square)

∠ABC=∠BAD ( = 90° )

ΔABC≅ΔBAD( By SAS property)

AC=BD ( by CPCT).

(ii) In a ΔOAD and ΔOCB,

AD=CB ( opposite sides of a square)

∠OAD=∠OCB ( transversal AC )

∠ODA=∠OBC ( transversal BD )

ΔOAD≅ΔOCB (ASA property)

OA=OC ---------(i)

Similarly OB=OD ----------(ii)

From (i) and (ii) AC and BD bisect each other.

Now in a ΔOBA and ΔODA,

OB=OD ( from (ii) )

BA=DA

OA=OA ( common line )

ΔAOB=ΔAOD----(iii) ( by CPCT

∠AOB+∠AOD=180° (linear pair)

2∠AOB=180°

∠AOB=∠AOD=90°

∴AC and BD bisect each other at right angles.

please mark me as BRAINLEST