Question

select the correct location on the table.
dana and sean are considering the quadrilateral with coordinates ( q(-2,4) ), ( r(1,5) ), ( s(2,2) ), and ( t(-1,1) ).
they already determined the sides of the quadrilateral are perpendicular and agree that qrst is a rectangle. however, sean believes qrst is also a square.
which set of work correctly determines whether sean is correct?
( qs=sqrt{(2 + 2)^{2}+(2 - 4)^{2}}=sqrt{20} )
( rt=sqrt{(-1 - 1)^{2}+(1 - 5)^{2}}=sqrt{20} )
therefore, qrst is a square.
( qr=sqrt{(1 + 2)^{2}+(5 - 4)^{2}}=sqrt{10} )
( rs=sqrt{(2 - 1)^{2}+(2 - 5)^{2}}=sqrt{10} )
therefore, qrst is a square.
( qt=sqrt{(-1 - 2)^{2}+(1 - 4)^{2}}=sqrt{18} )
( st=sqrt{(2 + 1)^{2}+(2 - 1)^{2}}=sqrt{10} )
therefore, qrst is a square.
( qs=sqrt{(2 + 2)^{2}+(2 - 4)^{2}}=sqrt{20} )
( rt=sqrt{(-1 - 1)^{2}+(1 - 5)^{2}}=sqrt{20} )
therefore, qrst is not a square.
( qr=sqrt{(1 + 2)^{2}+(5 - 4)^{2}}=sqrt{10} )
( rs=sqrt{(2 - 1)^{2}+(2 - 5)^{2}}=sqrt{10} )
therefore, qrst is not a square.
( qt=sqrt{(-1 - 2)^{2}+(1 - 4)^{2}}=sqrt{18} )
( st=sqrt{(2 + 1)^{2}+(2 - 1)^{2}}=sqrt{10} )
therefore, qrst is not a square.

Explanation:

Step1: Recall the properties of a square

A square is a rectangle with all sides equal.

Step2: Calculate the lengths of adjacent sides

For a quadrilateral \(QRST\) with vertices \(Q(- 2,4)\), \(R(1,5)\), \(S(2,2)\), \(T(-1,1)\)
The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
For side \(QR\):
\(QR=\sqrt{(1+2)^2+(5 - 4)^2}=\sqrt{9 + 1}=\sqrt{10}\)
For side \(RS\):
\(RS=\sqrt{(2 - 1)^2+(2 - 5)^2}=\sqrt{1+9}=\sqrt{10}\)
For side \(ST\):
\(ST=\sqrt{(2 + 1)^2+(2 - 1)^2}=\sqrt{9+1}=\sqrt{10}\)
For side \(TQ\):
\(TQ=\sqrt{(-1 + 2)^2+(1 - 4)^2}=\sqrt{1 + 9}=\sqrt{10}\)

Since all sides \(QR = RS=ST=TQ=\sqrt{10}\) and the sides are perpendicular (given that it is a rectangle), \(QRST\) is a square.

Answer:

The set of work \(QR=\sqrt{(1 + 2)^2+(5 - 4)^2}=\sqrt{10}\), \(RS=\sqrt{(2 - 1)^2+(2 - 5)^2}=\sqrt{10}\), \(ST=\sqrt{(2 + 1)^2+(2 - 1)^2}=\sqrt{10}\), \(TQ=\sqrt{(-1 + 2)^2+(1 - 4)^2}=\sqrt{10}\) and stating "Therefore, \(QRST\) is a square" is correct.