Question

show that the points (2, 3) (3 ,4)(5 ,6) and (4, 5 )are the vertices of parallelogram​

Answer 1

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Solution

Answer 2

Answer:  The proof is given below.

Step-by-step explanation:  We are given to show that the points (2, 3), (3 ,4), (5 ,6) and (4, 5) are the vertices of parallelogram​.

We know that

a quadrilateral is a parallelogram if one pair of opposite sides are congruent and parallel.

Let A, B, C and D denotes the vertices (2, 3), (3 ,4), (5 ,6) and (4, 5) respectively.

Then, the lengths of the sides AB and CD, as calculated using distance formula, are

$AB=\sqrt{(3-2)^2+(4-3)^2}=\sqrt{1+1}=\sqrt2,\\\\CD=\sqrt{(4-5)^2+(5-6)^2}=\sqrt{1+1}=\sqrt2.$

Also, the slopes of the sides AB and CD are

$\textup{slope of AB}=\dfrac{4-3}{3-2}=1,\\\\\\\textup{slope of CD}=\dfrac{5-6}{4-5}=1.$

Therefore, the lengths and slopes of sides AB and CD are equal. That is, one pair of opposite sides are congruent and parallel.

Thus, ABCD is a parallelogram.