Question

show that the points (1,1),( -1,5) ,(7,9), (9,5) takn in that order are the vertices of a rectangle

Answer 1

hope it helps........

Answer 2

Clearly, $PQ=RS$, $PS=QR$, and $PR=QS$

Hence, (1,1),( -1,5) ,(7,9), (9,5) taken in order are the vertices of a rectangle.

Explanation:

Given: Points (1,1),( -1,5) ,(7,9), (9,5) taken in order are the vertices of a rectangle.

To prove: These points are the vertices of a rectangle.

Proof:

Let the point (1,1) be P, ( -1,5) be Q, (7,9) be R, and (9,5) be S.

We know that distance between two points $(x_{1} ,y_{1} )$ and $(x_{2} ,y_{2} )$ is $\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$

Now,

$PQ=\sqrt{(-1-1)^{2}+(5-1)^{2} } =\sqrt{4+16} =\sqrt{20}$

$QR=\sqrt{(7+1)^{2}+(9-5)^{2} } =\sqrt{64+16} =\sqrt{80}$

$RS=\sqrt{(9-7)^{2}+(5-9)^{2} } =\sqrt{4+16} =\sqrt{20}$

$PS=\sqrt{(9-1)^{2}+(5-1)^{2} } =\sqrt{64+16} =\sqrt{80}$

$PR=\sqrt{(7-1)^{2}+(9-1)^{2} } =\sqrt{36+64} =10$

$QS=\sqrt{(9+1)^{2}+(5-5)^{2} } =\sqrt{100+0} =10$

Clearly, $PQ=RS$, $PS=QR$, and $PR=QS$

Hence, (1,1),( -1,5) ,(7,9), (9,5) taken in order are the vertices of a rectangle.

Learn more:

Distance between two points

https://brainly.in/question/7016568