Prove that the diagonals of a square are equal and perpendicular to each other.
Answers
since the diagonals makes an angular bisection, they divide the angles into 45 degrees.
or
when we plot a square in a graph, using mid point thearom, the mid points oflet both the diagonals should be the same ie. centroid of the square.
for example refer the given sum and hope you will understand.
let ABCD be a square, with the co ordinates (-2,-1),(1,0),(4,3),(1,2).
mid point of diagonal AC = (-2+4)/2,(-1+3)/2 = AC(1,1)
mid point of the diagonal BD (1+1)/2,(2+0)/2 =(1,1)
since AC=BD this can be a square.
Given :- ABCD is a square.
To proof :- AC = BD and AC ⊥ BD
Proof :- In △ ADB and △ BCA
AD = BC [ Sides of a square are equal ]
∠BAD = ∠ABC [ 90° each ]
AB = BA [ Common side ]
△ADB ≅ △BCA [ SAS congruency rule ]
⇒ AC = BD [ Corresponding parts of congruent triangles are equal ]
In △AOB and △AOD
OB = OD [ Square is also a parallelogram therefore, diagonal of parallelogram bisect each other ]
AB = AD [ Sides of a square are equal ]
AO = AO [ Common side ]
△AOB ≅ △ AOD [ SSS congruency rule ]
⇒ ∠AOB = ∠AOD [ Corresponding parts of congruent triangles are equal]
∠AOB + ∠AOD = 180° [ Linear pair ]
∠ AOB = ∠AOD = 90°
⇒ AO ⊥ BD
⇒ AC ⊥ BD
Hence proved, AC = BD and AC ⊥BD
