Ex 7.1, 5
In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. 7.8. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees. Using distance formula, find which of them is correct.
As seen from the figure, four points are
A(3, 4) , B(6, 7)
C(9, 4) , D(6, 1)
In a square,
All sides are equal
All diagonals are equal
Hence, we have to prove
AB = BC = CD = AD and AC = BD
Finding sides using Distance Formula
Finding AB
AB = √((𝑥2 −𝑥1)2+(𝑦2 −𝑦1)2)
= √(( 6 −3)2+(7 −4)2)
= √((3)2+(3)2)
= √(2(3)2)
= 3√𝟐
Finding BC
BC = √((𝑥2 −𝑥1)2+(𝑦2 −𝑦1)2)
= √(( 6 −3)2+(4 −7)2)
= √((3)2+(−3)2)
= √((3)2+(3)2)
= √(2(3)2)
= 3√𝟐
Finding CD
CD = √((𝑥2 −𝑥1)2+(𝑦2 −𝑦1)2)
= √(( 6 −9)2+(1 −4)2)
= √((−3)2+(−3)2)
= √((3)2+(3)2)
= √(2(3)2)
= 3√𝟐
Finding AD
AD = √((𝑥2 −𝑥1)2+(𝑦2 −𝑦1)2)
= √(( 6 −3)2+(1 −4)2)
= √((3)2+(−3)2)
= √((3)2+(3)2)
= √(2(3)2)
= 3√𝟐
Similarly finding the diagonals
Finding AC
AC = √((𝑥2 −𝑥1)2+(𝑦2 −𝑦1)2)
= √((9 −3)2+(4−4)2)
= √((6)2+(0)2)
= √((6)2)
= 6
Finding BD
BD = √((𝑥2 −𝑥1)2+(𝑦2 −𝑦1)2)
= √(( 6 −6)2+(1 −7)2)
= √((0)2+(−6)2)
= √((6)2)
= 6
Now since AB = BC = CD = AD = 3√2
& AC = AD = 6
Since the sides of ABCD are equal and diagonals are equal
So, ABCD is a square
Therefore, Champa is correct.
Now since AB = BC = CD = AD = 3√2
& AC = AD = 6
Since the sides of ABCD are equal and diagonals are equal
So, ABCD is a square
Therefore, Champa is correct.

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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