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

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

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Solution 1

Solution:

ΔADC and ΔBDC are right angled triangles with AD and BC as hypotaneus

`AC^2=BA^2+BC^2`

`AC^2=(5-2)^2+(6+1)^2=9+49=58 sq.unit`

`BD^2=DC^2+CB^2`

`BD^2=(5-2)^2+(-1-6)^2=9+49=58 sq.unit`

Hence, both the diagonals are equal in length.

Solution 2

The vertices of the rectangle ABCD are A(2, -1), B(5, -1), C(5, 6) and D(2, 6) Now,

`"Coordinates of midpoint of" AC = ((2+5)/2 , (-1+6)/2) = (7/5 ,5/2)`

`"Coordinates of midpoint of " BD = ((5+2)/2 , (-1+6)/2)= (7/2,5/2)`

Since, the midpoints of AC and BD coincide, therefore the diagonals of rectangle ABCD bisect each other.

Concept: Right-angled Triangles and Pythagoras Property
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