Question

Prove that the diagonals of a rectangle ABCD,with vertices A (2,-1),B (5,-1),C (5,6),D (2,6),are equal and bisect each ohter

Answer 1

We have ABCD as the given rectangle. in which AC and BD are the diagonals that intersect at O.
We will use distance formula to calculate the length of each diagonal.
Distance between 2 points  x 1 ,   y 1   and   x 2 ,   y 2  is given by,

distance   =   x 2   -   x 1 2   +   y 2   -   y 1 2         distance   formula 
Using the above formula, we get,

length   of   diagonal   AC   =   5 - 2 2   +   6 + 1 2   =   3 2   +   7 2   =   9   +   49   =   58   units length   of   diagonal   BD   =   5 - 2 2   +   - 1 - 6 2   =   3 2   +   - 7 2   =   9   +   49   =   58   unitsSo ,   diagonals   of   the   rectangle   are   equal . 
Now, to prove that, diagonals bisect each other, we will show that mid point of both the diagonals is same. For finding the mid point, we will use mid point formula.
Coordinates   of   the   mid   point   of   line   joining   points   x 1 ,   y 1   and   x 2 ,   y 2   is   given   by   x   =   x 1   +   x 2 2       ;     y   =   y 1   +   y 2 2 

Now coordinates of midpoint of AC are  2 + 5 2 ,   - 1   + 6 2   ≡ 7 2 ,   5 2         
       
coordinates of midpoint of BD are  2 + 5 2 ,   6 - 1 2 ≡ 7 2 ,   5 2 

Answer 2

Answer:

Step-by-step explanation:

We have ABCD as the given rectangle. in which AC and BD are the diagonals that intersect at O.

We will use distance formula to calculate the length of each diagonal.

Distance between 2 points  x 1 ,   y 1   and   x 2 ,   y 2  is given by,

distance   =   x 2   -   x 1 2   +   y 2   -   y 1 2         distance   formula 

Using the above formula, we get,

length   of   diagonal   AC   =   5 - 2 2   +   6 + 1 2   =   3 2   +   7 2   =   9   +   49   =   58   units length   of   diagonal   BD   =   5 - 2 2   +   - 1 - 6 2   =   3 2   +   - 7 2   =   9   +   49   =   58   unitsSo ,   diagonals   of   the   rectangle   are   equal . 

Now, to prove that, diagonals bisect each other, we will show that mid point of both the diagonals is same. For finding the mid point, we will use mid point formula.

Coordinates   of   the   mid   point   of   line   joining   points   x 1 ,   y 1   and   x 2 ,   y 2   is   given   by   x   =   x 1   +   x 2 2       ;     y   =   y 1   +   y 2 2 

Now coordinates of midpoint of AC are  2 + 5 2 ,   - 1   + 6 2   ≡ 7 2 ,   5 2         

      

coordinates of midpoint of BD are  2 + 5 2 ,   6 - 1 2 ≡ 7 2 ,   5 2 

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