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Parallelogram, Rhombus, Square, Rectangle

How to show that A, B, C, D are the vertices of a parallelogram?

Step 1
Show that the diagonals AC and BD bisect each other.


How will you show that A, B, C, D are the vertices of a Rhombus?

Step 1
Show that the diagonals AC and BD bisect each other. (That is, A, B, C, D forms a parallelogram)

Step 2 
Show that a pair of adjacent sides, for instance, AB and BC are equal.

 

How will you show that A, B, C, D are the vertices of a Square?

Step 1
Show that the diagonals AC and BD bisect each other.

Step 2
Show that two adjacent sides, for instance, AB and BC are equal.

Step 3
Show that the two diagonals AC and BD are equal.

 

How will you show that A, B, C, D are the vertices of a Rectangle?

Step 1
Show that the diagonals AC and BD bisect each other.

Step 2
Show that the two diagonals AC and BD are equal.

 

Example

Prove that the points A(−2, −1), B(1, 0), C(4, 3) and D(1, 2) are the vertices of a parallelogram.

Solution

Note that to show that a quadrilateral is a parallelogram, it is sufficient to show that the diagonals of the quadrilateral bisect each other.

Let M be the mid-point of AC, then coordinates of M are given by

Let N be the mid-points of BD, then coordinates of N are

 

Thus the mid-point of AC is same as that of the mid-point of BD. In other words AC and BD bisect each other. Hence, ABCD is a parallelogram.


 

Example

Show that the points A(1, 3), B(2, 6), C(5, 7) and D(4, 4) are the vertices of a rhombus.

Solution

Note that to show that a quadrilateral ABCD is a rhombus, it is sufficient to show that

(a) ABCD is a parallelogram, that is AC and BD have the same mid-point and

(b) a pair of adjacent edges are equal, for instance AB = BC.

In present case, mid-point of AC is and mid-point of BD is

Thus ABCD is a parallelogram.

Also, AB2 = (2 – 1)2 + (6 – 3)2 = 1 + 9 = 10

and BC2 = (5 – 2)2 + (7 – 6)2 = 9 +1 = 10

Therefore, AB2 = BC2 or AB = BC.

Thus, ABCD is a parallelogram in which a pair of adjacent edges are equal.

Hence ABCD is a rhombus.


 

Example

Show that the points A(3, 2), B(5, 4), C(3, 6) and D(1, 4) are the vertices of a square.

Solution

Note that to show that a quadrilateral is a square, it is sufficient to show that

(a) ABCD is a parallelogram, that is, AC and BD bisect each other,

(b) a pair of adjacent edges are equal, for instance AB = BC and

(c) the diagonal AC and BD are equal.

In the present case, the midpoint of AC is and mid-point of BD is

Therefore, ABCD is a parallelogram.

Also, AB2 = (5 – 3)2 + (4 – 2)2 = 4 + 4 = 8

  and BC2 = (3 – 5)2 + (6 – 4)2 = 4 + 4 = 8

Therefore, AB2 = BC2 or AB = BC.

Finally, AC2 = (3 – 3)2 + (6 – 2)2 = 16,

and BD2 = (1 – 5)2 + (4 – 4)2 = 16

That is, AC2 = BD2 or AC = BD. Hence, ABCD is a square.

 
Example

Show that the points A(1, −1), B(−2, 2), C(4, 8) and D(7, 5) are the vertices of a rectangle.

Solution

To show that ABCD is a rectangle, it is sufficient to show that

(a) ABCD is a parallelogram, that is, AC and BD bisect each other, and

(b) the diagonal AC and BD are equal.

The mid-point of AC is and mid-point of BD is Therefore, ABCD is a parallelogram.

Also, AC2 = (4 – 1)2 + (8 + 1)2 = 9 + 81 = 90

and BD2 = (7 + 2)2 + (5 – 2)2 = 81 + 9 = 90.

Thus, AC2 = BD2 or AC = BD.

Hence, ABCD is a rectangle.






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