## Show that the points (4, 6), (-1, 5), (-2, 0), (3, 1) are the vertices of a rhombus.​

Question

Show that the points (4, 6), (-1, 5), (-2, 0), (3,
1) are the vertices of a rhombus.​

in progress 0
1 month 2021-09-09T09:00:58+00:00 1 Answer 0

|AB|=|BC|=|CD|=|DA|=√26

This shows that the points are vertices of a rhombus.

Step-by-step explanation:

Show that the points (4, 6), (-1, 5), (-2, 0), (3,
1) are the vertices of a rhombus, we simply have to show that they all have equal distances.

Let ABCD be the vertices of the rhombus (4, 6), (-1, 5), (-2, 0), (3,
1)  respectively.  That is;

A(4, 6), B(-1, 5), C(-2, 0), D(3,
1)

We will find the distance AB, BC, CD and DA, If the distances between them are equal, then we are able to prove that it is a rhombus;

Using the distance formula;

D = √(  )² + (  Distance AB

A(4, 6), B(-1, 5) =4 =6 =-1 =5

|AB|=√(  )² + (  =√(5- 6)² + (-1- 4)²

=√(-1)² + (-5)²

=√1+ 25

=√26

Distance BC

B(-1, 5), C(-2, 0) =-1 =5 =-2 =0

|BC|=√(  )² + (  =√(0 – 5)² + (-2+1)²

=√(-5)² + (-1)²

=√25+1

=√26

Distance CD

C(-2, 0), D(3,
1) =-2 =0 =3 =1

|CD|=√(  )² + (  =√(1 – 0)² + (3+2)²

=√(1)² + (5)²

=√1+25

=√26

Distance DA

D(3,
1)  A(4,6) =3 =1 =4 =6

|DA|=√(  )² + (  =√(6 – 1)² + (4-3)²

=√(5)² + (1)²

=√25+1

=√26

|AB|=|BC|=|CD|=|DA|=√26

This shows that the points are vertices of a rhombus.