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Question

Prove that the diagonals of a rectangle bisect each other and are equal.


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Solution

Concept:
Application:

Let OACB be a rectangle such that OA is along x-axis and OB is along y-axis. Let OA = a and OB = b.

Then, the coordinates of A and B are (a,0) and (0, b) respectively

Since, OACB is a rectangle. Therefore, AC=OBAC=b

Thus, we have

OA = a and AC = b

So, the coordinates of C are (a, b)

The coordinates of the mid-points of OC are (a+02,b+02)=a2,b2

Also, the coordinates of the mid-points of AB are (a+02,0+b2)=a2,b2

Clearly, coordinates of the mid-point of OC and AB are same.

Hence, OC and AB bisect each other

Also, OC=a2+b2 and AB=(a0)2+(0b)2=a2+b2

OC=AB


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