Question

In a classroom, 4 friends are seated at the points A, B, C and D as shown in the given figure. 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.​

Answer 1

Question :-

In a classroom, 4 friends are seated at the points A, B, C and D as shown in the given figure. 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.

Solution :-

Here, we observe the rows and columns in that case we will see four coordinates which name is ABCD. And we need to find ABCD are the square or not.

Taking coordinates of ABCD,

From the figure co-ordinates of points can be expressed as shown below,

A(3,4), B(6,7), C(9,4), D(6,1)

Now,

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

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

$\rm{ CD= \sqrt{(9 - 6) { }^{2} + (4 - 1) {}^{2} } } \\ \\ \rm{ = \sqrt{3 {}^{2} + 3 {}^{2} } } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm{ = \sqrt{9 + 9} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm{ = \sqrt{18} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm{ =3 \sqrt{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\rm{ DA= \sqrt{(6 - 3) {}^{2} + (4 - 1) {}^{2} } } \\ \\ \rm{ = \sqrt{3 {}^{2} + 3 {}^{2} } } \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm{ = \sqrt{ 9 + 9} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm{ = \sqrt{18} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm{ = 3 \sqrt{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Four sides of the quadrilateral ABCD are equal.

$\rm\hookrightarrow{AB=BC=CD=DA=3 \sqrt{2} }$

Now, we see diagonals

$\rm:\longmapsto{AC_{\small{(Diagonal)}} = \sqrt{(9 - 3) {}^{2} + (4 - 4) {}^{2} } } \\ \\ \rm:\longmapsto{AC_{\small{(Diagonal)}} = \sqrt{6 {}^{2} } } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bf:\longmapsto \red{AC_{\small{(Diagonal)}} =6 } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\rm:\longmapsto{BD_{\small{(Diagonal)}} = \sqrt{(6 - 6) {}^{2} + (1 - 7) {}^{2} } } \\ \\ \rm:\longmapsto{BD_{\small{(Diagonal)}} = \sqrt{ - 6 {}^{2} } } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bf:\longmapsto \red{BD_{\small{(Diagonal)}} =6 } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\rm{\therefore \boxed{ \rm{\: Diagonal \: AC = Diagonal \: BD}}}$

Hence,

• ABCD is a square.

And,

• Champa is correct between two.

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In a classroom, 4 friends are seated at the points A, B, C and D as shown in Figure. Jarina and Phani walk into the class and after observing for a few minutes Jarina asks phani “Don’t you think ABCD is a square?” Phani disagrees. Using distance formula, find which of them is correct. Why?

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