Question

Find the area of the square formed by points (1,7) (4,2) (-1,-1) and (-4,4) please solve this no scam I need help please solve this fast. please​

Answer 1

$\large\underline\purple{\bold{Solution :- }}$

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Let the coordinates be represented as

A(1,7), B(4,2), C(-1,-1) and D(-4,4)

Since, ABCD is a square.

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There are two methods to find the area of square.

Method :- 1.

• Using Distance Formula

Method :- 2

• Using area of triangle

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Using Distance Formula

We know,

Distance between two points A and B is given by

$\bf\implies \:AB = \sqrt{ {(x_2-x_1)}^{2} + {(y_2-y_1)}^{2} }$

where coordinates of A and B are

$\tt \: A(x_1, \: y_1) \: and \: B(x_2, \: y_2)$

So, Distance between A(1, 7) and B(4, 2) is

$\bf\implies \:AB = \sqrt{ {(4 - 1)}^{2} + {(2 - 7)}^{2} }$

$\bf\implies \:AB = \sqrt{ {(3)}^{2} + {( - 5)}^{2} }$

$\bf\implies \:AB = \: \sqrt{9 + 25}$

$\bf\implies \:AB = \: \sqrt{34} \: units$

Now,

We know,

$\bf \: Area \: of \: square \: = {(side)}^{2}$

$\bf \: Area \: of \: square \: = {(AB)}^{2}$

$\bf \: Area \: of \: square \: = {( \sqrt{34} )}^{2}$

$\boxed{ \blue{ \bf \: Area \: of \: square \: = \: 34 \: sq. \: units}}$

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Using Area of Triangle

Let us find the area of triangle ABC

We know,

$\bf \ Area =\dfrac{1}{2} [x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]$

where

$\begin{gathered}\begin{gathered} \sf {(x_{1} , y_{1} )=(1,7)}\end{gathered}\end{gathered} \\ \begin{gathered}\begin{gathered} \sf {(x_{2} , y_{2} )=(4,2)}\end{gathered}\end{gathered} \\ \begin{gathered}\begin{gathered} \sf {(x_{3} , y_{3} )=( - 1, - 1)}\end{gathered}\end{gathered}$

So, area of triangle ABC is

$\tt \: Area_{( \triangle \: ABC)}\: = \dfrac{1}{2} \{1(2 + 1) + 4( - 1 - 7) - 1(7 - 2) \}$

$\tt \: Area_{( \triangle \: ABC)}\: = \dfrac{1}{2} |3 - 32 - 5|$

$\tt \: Area_{( \triangle \: ABC)}\: = \dfrac{1}{2} |3 - 37|$

$\tt \: Area_{( \triangle \: ABC)}\: = \dfrac{1}{2} | - 34|$

$\boxed{ \purple{ \tt \: Area_{( \triangle \: ABC)}\: = 17 \: sq. \: units}}$

So, Area of square ABCD is given by

$\bf \: Area \: of \: square \: = 2 \times \tt \: Area_{( \triangle \: ABC)}$

$\bf \: Area \: of \: square \: = 2 \times 17$

$\bf \: Area \: of \: square \: = \: 34 \: sq. \: units$

$\boxed{ \blue{ \bf \: Area \: of \: square \: = \: 34 \: sq. \: units}}$