Mathematics: Prove a quadrilateral with vertices g(1,-1), h(5,1), i(4,3) and j(0,1) is a rectangle., using rectangle method 1.

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Prove a quadrilateral with vertices g(1,-1), h(5,1),

tradee001

09.03.2022

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°.

$GH^{2} =[1-(-1)\^]{2} +(5-1)^{2}$

$=2^{2} +4^{2}$

= 20

$IJ^{2} =(1-3)^{2} +(0-4)^{2}$

= 20

Therefore, GH = IJ

$HI^{2} =(3-1)^{2} +(4-5)^{2}$

$= 2^{2} +1^{2}$

= 5

$GJ^{2} =[1-(-1)]^{2} +(0-1)^{2}$

$= 2^{2} +1^{2}$

= 5

Therefore, HI = GJ

Two pairs of opposite sides are equal and hence GHIJ is a parallelogram.

Now, in Δ GHI,

$GH^{2} =20$

$HI^{2} =5$

$GI^{2} =[3-(-1)]^{2} +(4-1)^{2}$

$=4^{2} +3^{2}$

= 25

Therefore, $GI^{2} =GH^{2} +HI^{2}$ .

This shows that Δ GHI is a right angled triangle and ∠ H = 90°.

Hence, GHIJ is a rectangle.

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