Question

Show that the points A(1,1), b(-1,5) ,C(7,9 and D(9,5) are the vertices of a rectangle ABCD

Answer 1

Answer:

opposite sides are equal and diagonals are equal.

The given points are the vertices of a rectangle

Step-by-step explanation:

It is given that, A(1,1), B(-1,5) ,C(7,9 and D(9,5) are points

Let (x1,y1 ) and (x2,y2) are the two end points. the length of the line is given by

$\sqrt{(x1-x2)^{2}-(y2-y1)^{2} }$

To find the length of AB

A(1,1), B(-1,5)

$AB=\sqrt{(-1-1)^{2}-(5-1)^{2} }$

AB=√20

To find the length of BC

B(-1,5),C(7,9)

$BC=\sqrt{(7--1)^{2}-(9-5)^{2} }$

BC=√80

To find the length of CD

C(7,9) D(9,5)

$CD=\sqrt{(9-7)^{2}-(5-9)^{2} }$

CD=√20

To find the length of AD

A(1,1), D(9,5)

$AD=\sqrt{(9-1)^{2}-(5-1)^{2} }$

AB=√80

To find diagonals AC and BD

$AC=\sqrt{(7-1)^{2}-(9-1)^{2} }$

AC=√100= 10

$BD=\sqrt{(9--1)^{2}-(5-5)^{2} }$

BD=√100= 10

AB =CD and BC = AD(opposit sides are equlal)

AC = BD (diagonals are equal)

Therefore ABCD is a rectangle

Answer 2

It is given that, A(1,1), B(-1,5) ,C(7,9 and D(9,5) are points

Let (x1,y1 ) and (x2,y2) are the two end points. the length of the line is given by

\sqrt{(x1-x2)^{2}-(y2-y1)^{2} }(x1−x2)2−(y2−y1)2

To find the length of AB

A(1,1), B(-1,5)

AB=\sqrt{(-1-1)^{2}-(5-1)^{2} }AB=(−1−1)2−(5−1)2

AB=√20

To find the length of BC

B(-1,5),C(7,9)

BC=\sqrt{(7--1)^{2}-(9-5)^{2} }BC=(7−−1)2−(9−5)2

BC=√80

To find the length of CD

C(7,9) D(9,5)

CD=\sqrt{(9-7)^{2}-(5-9)^{2} }CD=(9−7)2−(5−9)2

CD=√20

To find the length of AD

A(1,1), D(9,5)

AD=\sqrt{(9-1)^{2}-(5-1)^{2} }AD=(9−1)2−(5−1)2

AB=√80

To find diagonals AC and BD

AC=\sqrt{(7-1)^{2}-(9-1)^{2} }AC=(7−1)2−(9−1)2

AC=√100= 10

BD=\sqrt{(9--1)^{2}-(5-5)^{2} }BD=(9−−1)2−(5−5)2

BD=√100= 10

AB =CD and BC = AD(opposit sides are equlal)

AC = BD (diagonals are equal)

Therefore ABCD is a rectangle