Question

The points (−2, 1), (0, 3), (2, 1) and (0, −1) are the vertices of a ____.
A) Parallelogram
B) Scalene quadrilateral
C) Isosceles quadrilateral
D) Concave quadrilateral

Answer 1

Hint: For solving this question, we use the concept that if the points are \[P\left( {{x}_{1}},{{y}_{1}} \right)\] and \[Q\left( {{x}_{2}},\text{ }{{y}_{2}} \right),\] then the slope of the line joining PQ is $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$. Four different points are given in the question. Using these points, we calculate slopes of 4 lines. Using these slopes, we can easily determine the shape of the figure.
Complete step-by-step answer:
If the points are \[P\left( {{x}_{1}},{{y}_{1}} \right)\] and \[Q\left( {{x}_{2}},\text{ }{{y}_{2}} \right),\] the slope of the line joining PQ is $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$.
Points given in the questions are A (-2, 1), B (0, 3), C (2, 1) and D (0, -1).
Now, slope of the line AB joining points A (-2, 1) and B (0, 3) is:
$\begin{align}
  & =\dfrac{3-1}{0-\left( -2 \right)} \\
 & =\dfrac{2}{0+2} \\
 & =1 \\
\end{align}$
Slope of line AB $=1.$
Slope of the line BC joining points B (0, 3) and C (2, 1) is:
$\begin{align}
  & =\dfrac{1-3}{2-0} \\
 & =\dfrac{-2}{2} \\
 & =-1 \\
\end{align}$
Slope of line AB $=-1.$
slope of the line CD joining points C (2, 1) and D (0, -1) is:
$\begin{align}
  & =\dfrac{-1-1}{0-2} \\
 & =\dfrac{-2}{-2} \\
 & =1 \\
\end{align}$
Slope of line CD $=1.$
Slope of the line DA joining points D (0, -1) and A (-2, 1) is:
$\begin{align}
  & =\dfrac{-1-1}{0-\left( -2 \right)} \\
 & =\dfrac{-2}{0+2} \\
 & =-1 \\
\end{align}$
Slope of line DA $=-1.$
As we can see that the slopes of AB and CD are the same. Therefore, AB||CD.
Similarly, the slope of BC and slope of DA are also the same. Therefore, BC||DA.
Since, the opposite sides are parallel to each other.
Hence, A, B, C and D are the vertices of the parallelogram.
Therefore, option (a) is correct.

Note: Students can also solve this problem by constructing a proper figure using the cartesian system of coordinates as shown above. Once the figure ABCD is obtained, we can easily predict that the figure is rhombus which is a special form of parallelogram.