Question

Question 6 (iii)
Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:
(iii) (4, 5), (7, 6), (4, 3), (1, 2)

Answer 1

(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.
Distance between the points is given by
$\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2}$
$\therefore AB=\sqrt{(4-7{)}^2+(5-6{)}^2}=\sqrt{(-3{)}^2+(-1{)}^2}$
$=\sqrt{9+1}=\sqrt{10}$
$BC=\sqrt{(7-4{)}^2+(6-3{)}^2}=\sqrt{(3{)}^2+(3{)}^2}$
$=\sqrt{9+9}=\sqrt{18}$
$CD=\sqrt{(4-1{)}^2+(3-2{)}^2}=\sqrt{(3{)}^2+(1{)}^2}$
$=\sqrt{9+1}=\sqrt{10}$
$AD=\sqrt{(4-1{)}^2+(5-2{)}^2}=\sqrt{(3{)}^2+(3{)}^2}$
$=\sqrt{9+9}=\sqrt{18}$
Diagnol $AC=\sqrt{(4-4{)}^2+(5-3{)}^2}=\sqrt{(0{)}^2+(2{)}^2}$
$=\sqrt{0+4}=2$
Diagnol $BD=\sqrt{(7-1{)}^2+(6-2{)}^2}=\sqrt{(6{)}^2+(4{)}^2}$
$=\sqrt{36+16}=\sqrt{52}=2\sqrt{3}$

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