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

vpkz world May 04, 2022

Q1. Show that if the diagonals of a square are equal and bisect each other at right angles

OR

Q2. Show that the diagonals of a square are equal and bisect each other at right angles.

Given that $A B C D$ is a square.

To prove : $A C = B D$ and $A C$ and $B D$ bisect each other at right angles.

Proof:

(i) In a $Δ A B C$ and $Δ B A D$ ,

$A B = A B$ ( common line)

$B C = A D$ ( opppsite sides of a square)

$∠ A B C = ∠ B A D$ ( = 90° )

$Δ A B C ≅ Δ B A D$ ( By SAS property)

$A C = B D$ ( by CPCT).

(ii) In a $Δ O A D$ and $Δ O C B,$

$A D = C B$ ( opposite sides of a square)

$∠ O A D = ∠ O C B$ ( transversal AC )

$∠ O D A = ∠ O B C$ ( transversal BD )

$Δ O A D ≅ Δ O C B$ (ASA property)

$O A = O C$ ---------(i)

Similarly $O B = O D$ ----------(ii)

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

Now in a $Δ O B A$ and $Δ O D A$ ,

$O B = O D$ ( from (ii) )

$B A = D A$

$O A = O A$ ( common line )

$Δ A O B = Δ A O D$ ----(iii) ( by CPCT

$∠ A O B + ∠ A O D = 180 °$ (linear pair)

$2 ∠ A O B = 180 °$

$∠ A O B = ∠ A O D = 90 °$

∴ $A C$ and $B D$ bisect each other at right angles.

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