Question

show that the diagonal of a square are equal and bisect each other at right angles​

Answer 1

Answer:

Step-by-step explanation:

Show that the diagonals of a square are equal and bisect each other at right angles.

Answer:

Let ABCD be a square. Let the diagonals AC and BD intersect each other at a point O. To prove

that the diagonals of a square are equal and bisect each other at right angles, we have to

prove AC = BD, OA = OC, OB = OD, and ∠AOB = 90º.

In ΔABC and ΔDCB,

AB = DC                   (Sides of a square are equal to each other)

∠ABC = ∠DCB        (All interior angles are of 90)

BC = CB                   (Common side)

So, ΔABC ≅ ΔDCB      (By SAS congruency)

Hence, AC = DB          (By CPCT)

Hence, the diagonals of a square are equal in length.

In ΔAOB and ΔCOD,

∠AOB = ∠COD          (Vertically opposite angles)

∠ABO = ∠CDO          (Alternate interior angles)

AB = CD                     (Sides of a square are always equal)

So, ΔAOB ≅ ΔCOD  (By AAS congruence rule)

Hence, AO = CO and OB = OD     (By CPCT)

Hence, the diagonals of a square bisect each other.

In ΔAOB and ΔCOB,

As we had proved that diagonals bisect each other, therefore,

AO = CO

AB = CB         (Sides of a square are equal)

BO = BO        (Common)

So, ΔAOB ≅ ΔCOB       (By SSS congruency)

Hence, ∠AOB = ∠COB      (By CPCT)

However, ∠AOB + ∠COB = 1800        (Linear pair)

2∠AOB = 1800

∠AOB = 900

Hence, the diagonals of a square bisect each other at right angles.

Answer 2

Figure:-

Solutions:-

• Let ABCD be a square.
• Let the diagonals AC and BD intersection each other at a point O.

To Prove that.

• The diagonals of a square are equal and bisect each other at right angles.

To prove

• AC = BD, OA = OC, OB = OD and <AOB = 90°

In ∆ABC and ∆DCB,

AB = DC (side of a square are equal other)

<ABC = <DCB (All interior angle are of 90°)

BC = CB (common side)

.:. ∆ABC ~ ∆DCD (By SAS Congruence)

.:. AC = DB (By CPCT)

Hence, the diagonals of a square are equal in length.

<AOB = <COD (Vertical opposite angle)

<ABO = <CDO (Alternate interior angle)

AB = CD (sides of a square are alswer equal)

.:. ∆AOB ~ ∆COD (By AAS congruence rule)

.:. AO = CO and OB = OD (By CPCT)

Hence, the diagonals of a square bisect each other.

In ∆AOB and ∆COB,

Proved that:-

• Diagonal bisect each other,

therefore,

AO = CO

AB = CB (sides of a square are equal)

BO = BO (common)

.:. ∆AOB ~ ∆COD (By SSS Congruence)

<AOB = <COD (By CPCT)

<AOB + <COB = 180° (Linear pair)

2<AOB = 180°

<AOB = 90°

Hence, the diagonals of a square bisect each other.