Consider the polygon in the xy-coordinate plane with vertices at points (1,,,0) and (3,-1). what is the most specific name for this polygon?

The name of the polygon is rectangle

Step-by-step explanation:

* Lets find the slope of the four sides and the length of two adjacent

 side to know what is the name of the figure

∵ The vertices are (1 , 3) , (3 , 4) , (5 , 0) , (3 , -1)

∵ The rule of the slope of a line which passes through the points

  (x1 , y1) and (x2 , y2) is m = (y2 - y1)/(x2 - x1)

- Let (x1 , y1) is (1 , 3) and (x2 , y2) is (3 , 4)

∴ m1 = (4 - 3)/(3 - 1) = 1/2

- Let (x1 , y1) is (3 , 4) and (x2 , y2) is (5 , 0)

∴ m2 = (0 - 4)/(5 - 3) = -4/2 = -2

- Let (x1 , y1) is (5 , 0) and (x2 , y2) is (3 , -1)

∴ m3 = (-1 - 0)/(3 - 5) = -1/-2 = 1/2

- Let (x1 , y1) is (3 , -1) and (x2 , y2) is (1 , 3)

∴ m4 = (3 - -1)/(1 - 3) = 4/-2 = -2

- The parallel lines have equal slopes

- The product of the slopes of the perpendicular lines is -1

∵ m1 = m3 and m2 = m4

∴ The sides contains points (1 , 3) , (3 , 4) and (5 , 0) , (3 , -1) are parallel

  and the sides contain points (3 , 4) , (5 , 0) and (3 , -1) , (1 , 3) are

  parallel

∵ m1 × m2 = 1/2 × -2 = -1

∴ The side contains points (1 , 3) , (3 , 4) is ⊥ to the line contains points

  (3 , 4) , (5 , 0)

∵ m3 × m4 = 1/2 × -2 = -1

∴ The side contains points (5 , 0) , (3 , -1) is ⊥ to the line contains points

  (3 , -1) , (1 , 3)

- Now lets find the length of the four sides by using the rule

 of the distance

- If a segment has two endpoints (x1 , y1) and (x2 , y2), then the length

 of the distance is √[(x2 - x1)² + (y2 - y1)²]

∵ The length of the side which contains points (1 , 3) and (3 , 4) is

 √[(3 - 1)² + (4 - 3)²] = √[4 + 1] = √5

∵ The length of the side which contains points (3 , 4) and (5 , 0) is

 √[(5 - 3)² + (0 - 4)²] = √[4 + 16] = √20 = 2√5

∵ The length of the side which contains points (5 , 0) and (3 , -1) is

 √[(3 - 5)² + (-1 - 0)²] = √[4 + 1] = √5

∵ The length of the side which contains points (3 , -1) and (1 , 3) is

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

∴ The four sides are not equal but each two opposite sides are equal

- From all above

# Each two opposite sides are parallel and equal

# Each two adjacent sides are perpendicular

∴ The name of the polygon is rectangle

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step-by-step explanation: