Prove that the points A (5, 6), B (2, 8), C (0, 6) and D (3, 4) form the vertices of a parallelogram. Is it a square?
Answers
Answer:
□ABCD is a parallelogram.
□ABCD is not a square.
Step-by-step-explanation:
We have given the coordinates of the points.
We have to prove that the points form a parallelogram.
A ≡ ( 5, 6 ) ≡ ( x₁, y₁ )
B ≡ ( 2, 8 ) ≡ ( x₂, y₂ )
C ≡ ( 0, 6 ) ≡ ( x₃, y₃ )
D ≡ ( 3, 4 ) ≡ ( x₄, y₄ )
We know that,
Slope of line = y₂ - y₁ / x₂ - x₁
Now,
Slope of line AB = y₂ - y₁ / x₂ - x₁
⇒ Slope of line AB = 8 - 6 / 2 - 5
⇒Slope of line AB = 2 / - 3
∴ Slope of line AB = - 2 / 3 - - ( 1 )
Now,
Slope of line BC = y₃ - y₂ / x₃ - x₂
⇒Slope of line BC = 6 - 8 / 0 - 2
⇒Slope of line BC = - 2 / - 2
⇒Slope of line BC = 2 / 2
∴ Slope of line BC = 1 - - ( 2 )
Now,
Slope of line CD = y₄ - y₃ / x₄ - x₃
⇒Slope of line CD = 4 - 6 / 3 - 0
⇒Slope of line CD = - 2 / 3
∴ Slope of line CD = - 2 / 3 - - ( 3 )
Now,
Slope of line AD = y₄ - y₁ / x₄ - x₁
⇒Slope of line AD = 4 - 6 / 3 - 5
⇒Slope of line AD = - 2 / - 2
⇒Slope of line AD = 2 / 2
∴ Slope of line AD = 1 - - ( 4 )
From ( 1 ) & ( 3 ),
∴ Slope of line AB = Slope of line CD
∴ Line AB || Line CD
From ( 2 ) & ( 4 ),
∴ Slope of line BC = Slope of line AD
∴ Line BC || Line AD
The opposite sides of the quadrilateral formed by the given points are parallel.
∴ □ABCD is a parallelogram.
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Now, by distance formula,
d ( A, B ) = √[ ( x₁ - x₂ )² + ( y₁ - y₂ )² ]
⇒ d ( A, B ) = √[ ( 5 - 2 )² + ( 6 - 8 )² ]
⇒ d ( A, B ) = √[ ( 3 )² + ( - 2 )² ]
⇒ d ( A, B ) = √( 9 + 4 )
⇒ d ( A, B ) = √13 - - ( 5 )
Now,
d ( A, D ) = √[ ( x₁ - x₃ )² + ( y₁ - y₃ )² ]
⇒ d ( A, D ) = √[ ( 5 - 0 )² + ( 6 - 6 )² ]
⇒ d ( A, D ) = √[ ( 5 )² + ( 0 )² ]
⇒ d ( A, D ) = √( 25 + 0 )
⇒ d ( A, D ) = √25
⇒ d ( A, D ) = 5 - - ( 6 )
From ( 5 ) & ( 6 )
∴ AB ≠ AD
The adjacent sides of the parallelogram are not equal.
∴ Parallelogram □ABCD is not a square.
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Additional Information:
1. Distance Formula:
The formula which is used to find the distance between two points using their coordinates is called distance formula.
- d ( A, B ) = √[ ( x₁ - x₂ )² + ( y₁ - y₂ )² ]
2. Section Formula:
The formula which is used to find the coordinates of the point which divides a line segment in a particular ratio is called section
formula.
- x = ( mx₂ + nx₁ ) / ( m + n )
- y = ( my₂ + ny₁ ) / ( m + n )