Question

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.

Answer 1

Steps explained below.

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.