Question

Name the type of quadrilateral formed, if any, by the following points:
(4, 5), (7, 6), (4, 3), (1, 2)

• A. Parallelogram
• B. Square
• C. Rectangle
• D. None of these

Answer 1

The correct option is A Parallelogram

(iii) Let the points (4, 5), (7, 6), (4, 3), and (1, 2) be representing the vertices A, B, C, and D of the given quadrilateral respectively.
$\therefore AB=\sqrt{(4-7{)}^2+(5-6{)}^2}=\sqrt{(-3{)}^2+(-1{)}^2}=\sqrt{9+1}=\sqrt{10}$
$BC=\sqrt{(7-4{)}^2+(6-3{)}^2}=\sqrt{(3{)}^2+(3{)}^2}=\sqrt{9+9}=\sqrt{18}$
$CD=\sqrt{(4-1{)}^2+(3-2{)}^2}=\sqrt{(3{)}^2+(1{)}^2}=\sqrt{9+1}=\sqrt{10}$
$AD=\sqrt{(4-1{)}^2+(5-2{)}^2}=\sqrt{(3{)}^2+(3{)}^2}=\sqrt{9+9}=\sqrt{18}$
Diagnol $AC=\sqrt{(4-4{)}^2+(5-3{)}^2}=\sqrt{(0{)}^2+(2{)}^2}=\sqrt{0+4}=2$
Diagnol $CD=\sqrt{(7-1{)}^2+(6-2{)}^2}=\sqrt{(6{)}^2+(4{)}^2}=\sqrt{36+16}=\sqrt{52}=2\sqrt{13}$

It can be observed that opposite sides of this quadrilateral are of the same length. However, the diagonals are of different lengths. Therefore, the given points are the vertices of a parallelogram.