Question

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

Answer 1

Step 1: Solve for the lengths of diagonals and sides

Let the given points be $A(4,5),B(7,6),C(4,3),D(1,2)$

Hence, the sides of the triangle will be $AB,BC,CD,AD$ respectively and $AC,BD$ will be the diagonals.

Calculating all the values by using distance formula $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$ we get

$AB=\sqrt{(7-4{)}^2+{(6-5)}^2}=\sqrt{3^2+1^2}=\sqrt{9+1}=\sqrt{10}$

$BC=\sqrt{(4-7{)}^2+{(3-6)}^2}=\sqrt{3^2+3^2}=\sqrt{9+9}=\sqrt{18}$

$CD=\sqrt{(1-4{)}^2+{(2-3)}^2}=\sqrt{3^2+1^2}=\sqrt{9+1}=\sqrt{10}$

$AD=\sqrt{(1-4{)}^2+{(2-5)}^2}=\sqrt{3^2+3^2}=\sqrt{9+9}=\sqrt{18}$

$AC=\sqrt{(4-4{)}^2+{(3-5)}^2}=\sqrt{0^2+2^2}=2$

$BD=\sqrt{(7-1{)}^2+{(6-2)}^2}=\sqrt{6^2+4^2}=\sqrt{36+16}=\sqrt{52}$

Step 2:Determine the type of the quadrilateral

We can observe that opposite pairs of sides of the quadrilateral are equal and the diagonals of the quadrilateral are not equal.

Only rectangle and parallelogram have pairs of opposite sides equal.

Only rhombus and parallelogram have unequal diagonals.

As parallelogram is common to both the properties, the given co-ordinates form a parallelogram.

Hence the given points form a parallelogram