Answer:
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
