Math, asked by shreyajindal11, 1 month ago


Prove that the points A(0, 1), B(1,4), C(4,3) and D (3,0) are the vertices of a
square.

Answers

Answered by ItzWhiteStorm
152

Answer:-

Given:-

  • The points of vertices of square are A(0,1),B (1,4),C(4,3) and D(3,0).

To find:-

  • Prove the points

Required formula:-

  • Distance between two points = √(x₂ - x₁)² + (y₂ - y₁)²

Here,Finding the distance between the points AB;

 \\  : \implies \sf{AB =  \sqrt{ {(x_2-x_1)}^{2} +  {(y_2-y_1)}^{2} } } \\  \\ : \implies \sf{AB = \sqrt{ {(1 - 0)}^{2}  +  {(4 - 1)}^{2} } } \\  \\ : \implies \sf{AB = { \sqrt{ {(1)}^{2} +  {(3)}^{2}  } }} \\  \\ : \implies \sf{AB =  \sqrt{1 + 9} } \\  \\ : \implies \underline {\boxed{\frak{AB =  \sqrt{10}\: units}}} \:  \red{ \bigstar} \\

Now, finding the distance between the points BC;

 \\ :\implies\sf{BC =  \sqrt{ {(4 - 1)}^{2} +  {(3 - 4)}^{2}}} \\  \\ :\implies\sf{BC =  \sqrt{ {(3)}^{2}  +  {( - 1)}^{2} } } \\  \\ :\implies\sf{BC =  \sqrt{9 + 1}} \\  \\ :\implies \underline{ \boxed{\frak{BC =  \sqrt{10}  \: units}}} \:  \green{ \bigstar} \\

finding the distance between the points CD;

 \\ :\implies\sf{CD = \sqrt{{(3-4)}^{2}+{(0-3)}^{2}}} \\  \\ :\implies\sf{CD = \sqrt{{(-1)}^{2}+{(-3)}^{2}}}\\ \\ :\implies\sf{CD = \sqrt{1+9}}\\ \\ :\implies\underline{\boxed{\frak{CD = \sqrt{10}\: units}}}  \: \blue{ \bigstar} \\

Finally,finding the distance between the points DA;

 \\ :\implies\sf{DA = \sqrt{{(0-3)}^{2}+{(1-0)}^{2}}}\\ \\ :\implies\sf{DA = \sqrt{{(-3)}^{2}+{(1)}^{2}}} \\  \\ :\implies\sf{DA = \sqrt{9 + 1}} \\  \\ :\implies\underline {\boxed{\frak{DA = \sqrt{10} \: units}}} \:  \pink{ \bigstar}\\

Therefore,

  • AB = BC = CD = DA are equal.

  • Hence,Proved.
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