Answer:
The coordinates of C(a,b).
The length of AC diagonal is equal to [tex]\sqrt{a^2+b^2}[/tex]
The length of BD diagonal is equal to [tex]\sqrt{a^2+b^2}[/tex].
Therefore, AC diagonal is congruent to BD diagonal.
Step-by-step explanation:
Given
ABCD is a rectangle.
AB=CD and BC=AD
[tex]m\angle A= m\angle B= m\angle C=m\angle D=90^{\circ}[/tex]
The coordinates of rectangle ABCD are A(0,0),B(a,0),C(a,b) and D(0,b).
Distance between two points [tex](x_1,y_1) [/tex] and [tex](x_2,y_2)[/tex] is given by the formula
=[tex]\sqrt{(x-2-x_1)^2+(y_2-y_1)^2}[/tex]
The distance between two points A (0,0) and C(a,b)
AC=[tex]\sqrt{( a-0)^2+(b-0)^2}[/tex]
AC= [tex]\sqrt{(a^2+b^2)}[/tex]
The length of AC diagonal is equal to [tex]\sqrt{(a^2+b^2)}[/tex].
Distance between the points B(a,0) and D(0,b)
BD=[tex]\sqrt{(0-a)^2+(b-0)^2}[/tex]
BD=[tex]\sqrt{(a^2+b^2)}[/tex]
The length of BD diagonal is equal to [tex]\sqrt{(a^2+b^2)}[/tex].
The diagonals of the rectangle have the same length.
Therefore, AC diagonal is congruent to BD diagonal.
Hence proved.
