Answer :
Answer:
Steps explained below.
Step-by-step explanation:
It is given that the vertices of the quadrilateral are G(1, -1), H(5, 1), I(4, 3) and J(0, 1).
A parallelogram is a rectangle if one of its angle is 90° (and therefore, all angles will be 90°).
A quadrilateral is a parallelogram if two pairs of opposite sides are equal.
So, lets prove GH = IJ, HI = GJ and H = 90°.
[tex]GH^{2} =[1-(-1)\^]{2} +(5-1)^{2}[/tex]
[tex]=2^{2} +4^{2}[/tex]
= 20
[tex]IJ^{2} =(1-3)^{2} +(0-4)^{2}[/tex]
= 20
Therefore, GH = IJ
[tex]HI^{2} =(3-1)^{2} +(4-5)^{2}[/tex]
[tex]= 2^{2} +1^{2}[/tex]
= 5
[tex]GJ^{2} =[1-(-1)]^{2} +(0-1)^{2}[/tex]
[tex]= 2^{2} +1^{2}[/tex]
= 5
Therefore, HI = GJ
Two pairs of opposite sides are equal and hence GHIJ is a parallelogram.
Now, in Δ GHI,
[tex]GH^{2} =20[/tex]
[tex]HI^{2} =5[/tex]
[tex]GI^{2} =[3-(-1)]^{2} +(4-1)^{2}[/tex]
[tex]=4^{2} +3^{2}[/tex]
= 25
Therefore, [tex]GI^{2} =GH^{2} +HI^{2}[/tex].
This shows that Δ GHI is a right angled triangle and ∠ H = 90°.
Hence, GHIJ is a rectangle.
