Question

Quadrilateral MIKE has vertices M(4,1), I(6,4), K(12,0), and E(10,-3). Use coordinate geometry to prove that quadrilateral MIKE is a rectangle.

Answer 1

Consider the figure,

The segment MI can be calculated as,

`MI=\sqrt[]{(4-1)^2+(6-4)^2}=\sqrt[]{13}`

The segment EK can be calculated as,

`EK=\sqrt[]{3^2+2^2}=\sqrt[]{13}`

The segment ME can be calculated as,

`ME=\sqrt[]{(-3-1)^2+(10-4)^2}=\sqrt[]{52}`

The segment IK can be calculated as,

`IK=\sqrt[]{(12-6)^2+(0-4)^2}=\sqrt[]{52}`

Thus, we have, opposite sides are equal and all the sides are not equal. Hence, we have, length are equal and breadths are equal, also length not equal to braedth. Therefore, the given quadrileteral is a rectangle.