Which statement explains how you could use coordinate geometry to prove that quadrilateral abcd is a square?
shape abcd is shown. point a is at negative 3, 5. point b is at 1, 7. point c is at 3, 3. point d is at negative 1, 1.
prove that all sides are congruent, and the slopes of consecutive sides are opposite reciprocals
prove that segments ad and ab are congruent and parallel
prove that opposite sides are congruent and that the slopes of consecutive sides are equal
prove that segments bc and cd are congruent and parallel

Prove that all sides are congruent, and the slopes of consecutive sides are opposite reciprocals ⇒ First answer

Step-by-step explanation:

* Lets revise the properties of the square

- In the square every two opposite sides are parallel

- In the square all sides are equal

- In the square the four angles are right angles

- In the square every two adjacent sides have opposite reciprocal slopes

- In the square the diagonals are equal and perpendicular

* So to prove the quadrilateral is a square you must have two conditions

 from the statements above

- Every to opposite sides parallel and two consecutive sides are equal

- The four sides are equal and each two consecutive sides are ⊥

- Its diagonals are equal and perpendicular

* Lets check which statement is right and prove it

# First

- Prove that all sides are congruent, and the slopes of consecutive

 sides are opposite reciprocals

∵ The meaning of the slopes of consecutive sides are opposite

   reciprocals is the consecutive sides are ⊥, because the slopes

   of the ⊥ segments are opposite reciprocal

- Lets prove this statement

- In quadrilateral ABCD

∵ A = (-3 , 5) , B = (1 , 7) , C = (3 , 3) , D = (-1 , 1)

∵ The length of the segment whose endpoints are (x1 , y1) , (x2 , y2)

  is √[(x2 - x1)² + (y2 - y1)²]

∴ AB = √[(1 - -3)² + (7 - 5)²] = √[(4)² + (2)²] = √[16 + 4] = √20

∴ BC = √[(3 - 1)² + (3 - 7)²] = √[(2)² + (-4)²] = √[4 + 16] = √20

∴ CD = √[(-1 - 3)² + (1 - 3)²] = √[(-4)² + (-2)²] = √[16 + 4] = √20

∴ AD = √[(-1 - -3)² + (1 - 5)²] = √[(2)² + (-4)²] = √[4 + 16] = √20

∴ The four sides are congruent ⇒ (1)

∵ The slope of the segment whose endpoints are (x1 , y1) , (x2 , y2)

   is

∴ The slope of

∴ The slope of

∴ The slope of

∴ The slope of

- From above

∵ The AB and BC are consecutive sides

∵ Their slopes are 1/2 , -2 ⇒ opposite reciprocal

∵ The BC and CD are consecutive sides

∵ Their slopes are -2 , 1/2 ⇒ opposite reciprocal

∵ The CD and DA are consecutive sides

∵ Their slopes are 1/2 , -2 ⇒ opposite reciprocal

∵ The DA and AB are consecutive sides

∵ Their slopes are -2 , 1/2 ⇒ opposite reciprocal

∴ The slopes of consecutive sides are opposite reciprocals ⇒ (2)

- From (1) and (2) the first statement is true

* Prove that all sides are congruent, and the slopes of consecutive

 sides are opposite reciprocals

answer: -8.2562234e+13

i believe it's a.

i'm not sure on this one, most likely correct.