Class 10

Math

All topics

Coordinate Geometry

In a classroom, $4$ friends are seated at the points $A$, $B$, $C$ and $D$ as shown in Fig. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, Dont you think $ABCD$ is a square? Chameli disagrees. Using distance formula, find which of them is correct. why?

If we join all these points then

By using distance formula

$=(x_{2}−x_{1})_{2}+(y_{2}−y_{1})_{2} $

$AB=(3−6)_{2}+(4−7)_{2} =(−3)_{2}+(−3)_{2} =9+9 =18 =32 $

$BC=(6−9)_{2}+(7−4)_{2} =(−3)_{2}+(3)_{2} =9+9 =18 =32 $

$CB=(9−6)_{2}+(4−1)_{2} =(3)_{2}+(3_{2}) =9+9 =18 =32 $

$AD=(3−6)_{3}+(4−1)_{2} =−3)_{2}+(3)_{2} =9+9 =18 =32 $

if we join the point $AC$ and $BD$ than

Diagonal$AC=(3−9)_{2}+(4−4)_{2} =(−6)_{2}+(0)_{2} =36 =6$

Diagonal $BD=(6−6)_{2}+(7−1)_{2} =(0)_{2}+(6)_{2} =36 =6$

Hence all the sides of a quadrilateral are the same and the diagonals are also the same so $ABCD$ is a square.

Hence Champa is correct.