show that (1,2), (-2,6), (5,8) and (8,4) are vertices of a parallelogram
Answers
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SOLUTION:-
➡️Let the given points as A (5,8), B (6,3), C(3,1) and D(2,6). By the definition of a parallelogram if the length of opposite sides will be equal, then it is parallelogram. So first let us find the length of all sides
Distance between two points = √(x₂ - x₁)² + (y₂ - y₁)²
➡️Length of AB
A (5,8) B (6,3)
x₁ = 5, y₁ = 8
x₂ = 6, y₂ = 3
length of AB = √(6 - 5)² + (3 - 8)²
= √(1)² + (-5)² ==> √1 + 25 = √26 -------- (1)
➡️Length of BC
B (6,3) C (3,1)
x₁ = 6 y₁ = 3
x₂ = 3 y₂ = 1
length of AB = √(3 - 6)² + (1 - 3)²
= √(-3)² + (-2)² ==> √9 + 4 = √13 -------- (2)
➡️Length of CD
C (3,1) D (2,6)
x₁ = 3 y₁ = 1
x₂ = 2 y₂ = 6
length of AB = √(2 - 3)² + (6 - 1)²
= √(-1)² + (5)² ==> √1 + 25 = √26 -------- (3)
➡️Length of DA
D (2,6) A (5,8)
x₁ = 2 y₁ = 6
x₂ = 5 y₂ = 8
length of AB = √(5 - 2)² + (8 - 6)²
= √(3)² + (2)² ==> √9 + 4 = √13 -------- (4)
➡️The opposite sides are having equal length.From this we can conclude that the given vertices form a parallelogram.
length of parallelogram = √26
width of parallelogram = √13
Perimeter of a parallelogram = 2 (L + W)
= 2 (√26 + √13) cm