Show that the points (2,3) (3,4) (5,6) and (4,5) are the vertices of a parallelogram.
Answers
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
\begin{gathered}AB=\sqrt{(3-2)^2+(4-3)^2}=\sqrt{1+1}=\sqrt2,\\\\CD=\sqrt{(4-5)^2+(5-6)^2}=\sqrt{1+1}=\sqrt2.\end{gathered}
AB=
(3−2)
2
+(4−3)
2
=
1+1
=
2
,
CD=
(4−5)
2
+(5−6)
2
=
1+1
=
2
.
Also, the slopes of the sides AB and CD are
\begin{gathered}\textup{slope of AB}=\dfrac{4-3}{3-2}=1,\\\\\\\textup{slope of CD}=\dfrac{5-6}{4-5}=1.\end{gathered}
slope of AB=
3−2
4−3
=1,
slope of CD=
4−5
5−6
=1.
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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
Answer:
Step-by-step explanation:
