Using the distributive property, an expression equivalent to 48+30

6(8×5)

Step-by-step explanation:

What I did was called factoring. To factor you find the Greatest Common Factor (GCF) of the numbers (in this case 6) and divide the GCF by the numbers (48/6=8 and 30/6=5).

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