Question

Name the type of quadrilateral formed by the following points A(3, 5) , B(6, 0) , C(1, - 3) and (- 2, 2) .

Answer 1

Answer:

Square

Step-by-step explanation:

Given,

A = (3 , 5)

B = (6 , 0)

C = (1 , - 3)

D = (-2 , 2)

To Find :-

The type of Quadrilateral

How To Do :-

First we need to find the length's of all sides using distance formula and if the lengths sides are equal we need to find the lengths of diagonals because there will be two possibilities 'square' and 'rhombus' if diagonal lengths are equal we can consider it as Square if they are not equal we can consider it as Rhombus.

Formula Required :-

Distance Formula :

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Solution :-

We need to find the lengths of  sides (AB , BC , CD , DA) , diagonals lengths(AC , BD).

First lets find the lengths of sides (AB , BC , CD , DA) :-

AB :-

A = (3 , 5)

let ,

x₁ = 3 , y₁ = 5

B = (6 , 0)

x₂ = 6 , y₂ = 0

Substituting in distance formula :-

$AB=\sqrt{(6-3)^2+(0-5)^2}$

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

$=\sqrt{9+25}$

∴ AB = √34

BC :-

B = (6 , 0)

x₁ = 6 , y₁ = 0

C = (1  -3)

x₂ = 1 , y₂ = - 3

Substituting in distance formula :-

$BC=\sqrt{(1-6)^2+(-3-0)^2}$

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

$=\sqrt{25+9}$

∴ BC = √34

CD :-

C = (1 , -3)

let,

x₁ = 1 , y₁ = - 3

D = (-2 , 2)

x₂ = - 2 , y₂ = 2

$CD=\sqrt{(-2-1)^2+(2-(-3))^2}$

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

$=\sqrt{9+25}$

∴ CD = √34

DA :-

D = (-2 , 2)

let,

x₁ = - 2 , y₁ = 2

A = (3 , 5)

x₂ = 3 , y₂ = 5

$DA=\sqrt{(3-(-2))^2+(5-2)^2}$

$=\sqrt{5^2+3^2}$

$=\sqrt{25+9}$

∴ DA = √34

We got all lengths are equal ( ∴ AB = BC = CD = DA)

Let's find the lengths of diagonals (AC , BD) :-

AC :-

A = (3 , 5)

let ,

x₁ = 3 , y₁ = 5

C = (1 , -3)

x₂ = 1 , y₂ = -3

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

$=\sqrt{(-2)^2+(-8)^2}$

$=\sqrt{4+64}$

∴ AC = √68

BD :-

B = (6 , 0)

x₁ = 6 , y₁ = 0

D = (-2  2)

x₂ = -2 , y₂ = 2

$BD=\sqrt{(-2-6)^2+(2-0)^2}$

$=\sqrt{(-8)^2+2^2}$

$=\sqrt{64+4}$

∴ BD = √68

We got Diagonal lengths are also equal(∴ AC = BD)

∴ Lengths of all sides and diagonals are equal we can say ABCD is 'square'