Question

Prove that the points A(2, -1), B(3, 4), C(-2, 3) and D(-3, -2) are the vertices of a rhombus ABCD. Is ABCD a square

Answer 1

Answer:

 ABCD is not a square

Step-by-step explanation:

• Gradient of AB = (4 - -1)/(3 - 2)  = 5/1  = 5
• Gradient of BC = (3 - 4)/(-2 - 3) = -1/-5 = 1/5
• Gradient of CD = (-2 - 3)/(-3 - -2) = -5/-1 = 5
• Gradient of DA = (-1 - -2)/(2 - -3) = 1/5

Since the gradients of AB and CD are equal, as are the gradients of BC and DA, it follows that ABCD is a parallelogram.

• Length of AB = √( (4 - -1)² + (3 - 2)² ) = √( 5² + 1² ) = √26
• Length of BC = √( (3 - 4)² + (-2 - 3)² ) = √( (-1)² + (-5)² ) = √26

Since ABCD is a parallelogram with AB=BC, it follows that ABCD is a rhombus.

Finally, two lines are perpendicular if the product of their gradients is equal to -1.  Use this to check if ABCD has right angles:

• Gradient of AB × gradient of BC  =  5 × 1/5  =  1  ≠  -1

So ABCD is not a square.

Answer 2

Answer:

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