Question

Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:

(i)$(-1,-2),(1,0),(-1,2),(-3,0)$

(ii) $(-3,5),(3,1),(0,3),(-1,-4)$

(iii) $(4,5),(7,6),(4,3),(1,2)$

Answer 1

Step 1: Name the quadrilateral$(-1,-2),(1,0),(-1,2),(-3,0)$

Let the points $(-1,-2),(1,0),(-1,2)\text{and}(-3,0)$ be represented by the vertices $\mathrm{A},\mathrm{B},\mathrm{C},$ and $\mathrm{D}$ of the given quadrilateral respectively.

Find the length of the side as follows:

$\begin{array}{c}\mathrm{AB}=\sqrt{[}(1+1{)}^2+(0+2{)}^2]\\ =\sqrt{[}4+4]\\ =2\sqrt{2}\end{array}$

$\begin{array}{c}\mathrm{BC}=\sqrt{[}(-1-1{)}^2+(2-0{)}^2]\\ =\sqrt{[}4+4]\\ =2\sqrt{2}\end{array}$

$\begin{array}{c}DA=\sqrt{[}(-3+1{)}^2+(0-2{)}^2]\\ =\sqrt{[}4+4]\\ =2\sqrt{2}\end{array}$

The length diagonal of the quadrilateral is calculated as follow:

$\begin{array}{c}AC=\sqrt{\left[(-1+1{)}^2+(2+2{)}^2\right]}\\ =\sqrt{[0}+16]\\ =4\end{array}$

$\begin{array}{c}BD=\sqrt{\left[(-3-1{)}^2+(0-0{)}^2\right]}\\ =\sqrt{[0}+16\\ =4\end{array}$

$\text{Side length}=\mathrm{AB}=\mathrm{BC}=\mathrm{CD}=\mathrm{DA}=2\sqrt{2}$

$\text{Diagonal Measure}=\mathrm{AC}=\mathrm{BD}=4$

Hence, the given points are the vertices of a square.

Step 2: Name the quadrilateral $(-3,5),(3,1),(0,3),(-1,-4)$

Let the points$(-3,5),(3,1),(0,3),(-1,-4)$ be represented by the vertices$\mathrm{A},\mathrm{B},\mathrm{C},$ and $\mathrm{D}$ of the given quadrilateral respectively.

Find the length of the side as follows:

$\begin{array}{c}\mathrm{AB}=\sqrt{\left[(-3-3{)}^2+(1-5{)}^2\right]}\\ =\sqrt{[36+16]}\\ =2\sqrt{13}\\ \mathrm{BC}=\sqrt{\left[\left(0-{13}^2+(3-1{)}^2\right]\right.}\\ =\sqrt{[9+4]}\\ =\sqrt{13}\\ \mathrm{CD}=\sqrt{\left[(-1-0{)}^2+(-4-3{)}^2\right]}\\ =\sqrt{[1+49]}\\ =5\sqrt{2}\\ \mathrm{DA}=\sqrt{\left[(-1+3{)}^2+(-4-5{)}^2\right]}\\ =\sqrt{[4+81]}\\ =\sqrt{85}\end{array}$

It is also observed that points $\mathrm{A},\mathrm{B}$ and $\mathrm{C}$ are collinear.

So, the given points can only form $3$ sides i.e, a triangle and not a quadrilateral that has $4$ sides.

Hence, the given points cannot form a general quadrilateral.

Step 3: Name the quadrilateral $(4,5),(7,6),(4,3),(1,2)$

Let the points $(4,5),(7,6),(4,3),(1,2)$ be represented by the vertices $A,B,C,$ and $D$ of the given quadrilateral respectively.

Find the length of the side as follows

:$\begin{array}{c}\mathrm{AB}=\sqrt{[}(7-4{)}^2+(6-5{)}^2]\\ =\sqrt{[}9+1]\\ =\sqrt{10}\\ \mathrm{BC}=\sqrt{[}\left(4-7^2+(3-6{)}^2\right]\\ =\sqrt{[9+9]}\\ =\sqrt{18}\\ \mathrm{CD}=\sqrt{[}(1-4{)}^2+(2-3{)}^2]\\ =\sqrt{[9+1]}\\ =\sqrt{10}\\ \mathrm{DA}=\sqrt{[}(1-4{)}^2+(2-5{)}^2]\\ =\sqrt{[9+9]}\\ =\sqrt{18}\end{array}$

The length diagonal of the quadrilateral is calculated as follow:

$\begin{array}{c}\mathrm{AC}=\sqrt{[}(4-4{)}^2+(3-5{)}^2]\\ =\sqrt{[0}+4]\\ =2\\ \mathrm{BD}=\sqrt{[}(1-7{)}^2+(2-6{)}^2]\\ =\sqrt{[}36+16]\\ =2\sqrt{13}\end{array}$

The opposite sides of this quadrilateral are of the same length. However, the diagonals are of different lengths.

Hence, the given points are the vertices of a parallelogram.