Question

Prove that four points whose coordinates are [0,5],[-2,-2],[5,0]and [7,7] form a rhombus.

Answer 1

To prove that it is a rhombus,

we will prove that all the 4 lengths formed by the 4 coordinates are equal.

Let the coordinates be A, B, C and D (as shown in attachment)

Find the length AB:

$\text {Length = } \sqrt{(Y_2 - Y_1)^2 + (X_2 - X_1)^2}$

Given that the coordinates are A(0,5) B(-2,-2)

$\text {Length = } \sqrt{(5+2)^2 + (0+2)^2}$

$\text {Length = } \sqrt{(7)^2 + (2)^2}$

$\text {Length = } \sqrt{53}$

Find the length BC:

$\text {Length = } \sqrt{(Y_2 - Y_1)^2 + (X_2 - X_1)^2}$

Given that the coordinates are B(-2,2) and C(5,0)

$\text {Length = } \sqrt{(0-2)^2 + (5+2)^2}$

$\text {Length = } \sqrt{(2)^2 + (7)^2}$

$\text {Length = } \sqrt{53}$

Find the length CD:

$\text {Length = } \sqrt{(Y_2 - Y_1)^2 + (X_2 - X_1)^2}$

Given that the coordinates are C(5,0) and D(7,7)

$\text {Length = } \sqrt{(0-7)^2 + (5-7)^2}$

$\text {Length = } \sqrt{(7)^2 + (2)^2}$

$\text {Length = } \sqrt{53}$

Find the length AD:

$\text {Length = } \sqrt{(Y_2 - Y_1)^2 + (X_2 - X_1)^2}$

Given that the coordinates are D(7,7) and A(0,5)

$\text {Length = } \sqrt{(7-5)^2 + (7-0)^2}$

$\text {Length = } \sqrt{(2)^2 + (7)^2}$

$\text {Length = } \sqrt{53}$

Since AB = BC = CD = AD

It is proven that ABCD is a rhombus

Answer 2

Step-by-step explanation:

i hope this will help you