Step-by-step explanation:

Using distance formula,

\sqrt{(x_{2}-x_{1} )^{2}+(y_{2}-y_{1} )^{2}}

(x

2

−x

1

)

2

+(y

2

−y

1

)

2

∴ PQ = \sqrt{(6-0)^{2}+(2+4)^{2}}

(6−0)

2

+(2+4)

2

=\sqrt{(6)^{2}+(6)^{2}} =\sqrt{36+36}=

(6)

2

+(6)

2

=

36+36

= \sqrt{72}

72

QR = \sqrt{(3-6)^{2}+(5-2)^{2}}

(3−6)

2

+(5−2)

2

=\sqrt{(-3)^{2}+(3)^{2}}=\sqrt{9+9}=\sqrt{18}=

(−3)

2

+(3)

2

=

9+9

=

18

RS = \sqrt{(-3-3)^{2}+(-1-5)^{2}}

(−3−3)

2

+(−1−5)

2

=\sqrt{(-6)^{2}+(-6)^{2}}=\sqrt{36+36} =\sqrt{72}=

(−6)

2

+(−6)

2

=

36+36

=

72

SP = \sqrt{(-3-0)^{2}+(-1+4)^{2}}

(−3−0)

2

+(−1+4)

2

=\sqrt{(-3)^{2}+(3)^{2}}=\sqrt{9+9} =\sqrt{18}=

(−3)

2

+(3)

2

=

9+9

=

18

∴ PQ = RS = \sqrt{72}

72

and

QR = SP = \sqrt{18}

18

Diagonals,

PR =\sqrt{(3-0)^{2}+(5+4)^{2}}

(3−0)

2

+(5+4)

2

=\sqrt{(3)^{2}+(9)^{2}}=\sqrt{9+81} =\sqrt{90}=

(3)

2

+(9)

2

=

9+81

=

90

QS = \sqrt{(-3-6)^{2}+(-1-2)^{2}}

(−3−6)

2

+(−1−2)

2

=\sqrt{(-9)^{2}+(-3)^{2}}=\sqrt{81+9} =\sqrt{90}=

(−9)

2

+(−3)

2

=

81+9

=

90

Diagonal PR = Diagonal QS = \sqrt{90}

90

, proved.

Thus, the points P (0, -4), Q (6, 2), R (3, 5) and S (-3, -1) are the vertices of a rectangle PQRS, proved.

Step-by-step explanation:

The four points P (0, - 4), Q (6, 2), R (3, 5) and S (- 3, - 1) are the vertices of a rectangle PQRS. Diagonal PR = Diagonal QS = , proved. Thus, the points P (0, -4), Q (6, 2), R (3, 5) and S (-3, -1) are the vertices of a rectangle PQRS, proved.

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