Answer
Verified
414.6k+ views
Hint: First we have to plot the given points into a graph. Because we must know the distance between all the four pairs of points. Then we will start with the use of the distance formula to find the distance between all the four pairs of the points. After calculating the vertex of the point $D$ then we will find the area of the square $ABCD$.
Formula used: Distance between points formula $D = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} $
Where $\left( {{x_1},{x_2}} \right)$ and $\left( {{y_1},{y_2}} \right)$ be the two points.
Area of the square $ = $ $side \times side$ unit sq.
Complete step-by-step answer:
Given points are $A = (3,4)$, $B = ( - 2,4)$, and $C = ( - 2, - 1)$
By plotting the points into graph we get,
First we have to find the fourth vertex $D$ and the area of the given square $ABCD$.
In the square $ABCD$, we know all the sides are equal thus, the distance of each of the vertices are same
Hence we can write,
$AB = BC = CD = DA$
The points plotted on the graph below.
From the graph, the coordinates of point $D$ are $(3, - 1)$
Hence,
Now using the distance formula $D = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} ......(1)$
Using equation $(1)$ we can find out the side $AB$, so it will be
Here, $A = (3,4)$ where ${x_1} = 3,{y_1} = 4$ and $B = ( - 2,4)$ where ${x_2} = - 2,{y_2} = 4$
$\Rightarrow$$AB = \sqrt {{{( - 2 - 3)}^2} + {{(4 - 4)}^2}} $ $ = \sqrt {{{( - 5)}^2} + {0^2}} $$ = \sqrt {( - 5)( - 5) + 0} $$ = \sqrt {25} $$ = 5$ Units
Here,
$B = ( - 2,4)$ where ${x_1} = - 2$, ${y_1} = 4$ and
$C = ( - 2, - 1)$ where ${x_2} = - 2,{y_2} = - 1$
$\Rightarrow$$BC = \sqrt {{{( - 2 - ( - 2))}^2} + {{( - 1 - 4)}^2}} $$ = \sqrt {0 + {{( - 5)}^2}} = \sqrt {25} = 5$ Units
Now,
$C = ( - 2, - 1)$ where ${x_1} = - 2,{y_1} = - 1$ and
$D = (3, - 1)$ where ${x_2} = 3,{y_2} = - 1$
$\Rightarrow$$CD = \sqrt {{{(3 - ( - 2))}^2} + {{( - 1 - ( - 1))}^2}} = \sqrt {{5^2} + {0^2}} $$ = \sqrt {25} = 5$ Units
Now,
$D = (3, - 1)$where ${x_1} = 3,{y_1} = - 1$ and
$A = (3,4)$where ${x_2} = 3,{y_2} = 4$
$\Rightarrow$$DA = \sqrt {{{(3 - 3)}^2} + {{( - 1 - 4)}^2}} = \sqrt {{0^2} + {{( - 5)}^2}} = \sqrt {25} = 5$ Units
$D$ are $(3, - 1)$
Here we got the side $AB = 5$ unit sq.
Area of the square $ = $ side ×side
Area of the square $ABCD = AB \times AB$
Substituting the side value,
$ \Rightarrow 5 \times 5$
Multiplying both the numbers, we get
Hence,
Area of the square \[ABCD = 25\] Unit sq.
$\therefore $ The area of the square is $25$ unit sq.
Note: Whenever we face such types of problems the key behind the solution lies in the diagrammatic representation of the data provided then awareness about the basic formula can help in getting to the right track to reach the answer.
Formula used: Distance between points formula $D = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} $
Where $\left( {{x_1},{x_2}} \right)$ and $\left( {{y_1},{y_2}} \right)$ be the two points.
Area of the square $ = $ $side \times side$ unit sq.
Complete step-by-step answer:
Given points are $A = (3,4)$, $B = ( - 2,4)$, and $C = ( - 2, - 1)$
By plotting the points into graph we get,
First we have to find the fourth vertex $D$ and the area of the given square $ABCD$.
In the square $ABCD$, we know all the sides are equal thus, the distance of each of the vertices are same
Hence we can write,
$AB = BC = CD = DA$
The points plotted on the graph below.
From the graph, the coordinates of point $D$ are $(3, - 1)$
Hence,
Now using the distance formula $D = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} ......(1)$
Using equation $(1)$ we can find out the side $AB$, so it will be
Here, $A = (3,4)$ where ${x_1} = 3,{y_1} = 4$ and $B = ( - 2,4)$ where ${x_2} = - 2,{y_2} = 4$
$\Rightarrow$$AB = \sqrt {{{( - 2 - 3)}^2} + {{(4 - 4)}^2}} $ $ = \sqrt {{{( - 5)}^2} + {0^2}} $$ = \sqrt {( - 5)( - 5) + 0} $$ = \sqrt {25} $$ = 5$ Units
Here,
$B = ( - 2,4)$ where ${x_1} = - 2$, ${y_1} = 4$ and
$C = ( - 2, - 1)$ where ${x_2} = - 2,{y_2} = - 1$
$\Rightarrow$$BC = \sqrt {{{( - 2 - ( - 2))}^2} + {{( - 1 - 4)}^2}} $$ = \sqrt {0 + {{( - 5)}^2}} = \sqrt {25} = 5$ Units
Now,
$C = ( - 2, - 1)$ where ${x_1} = - 2,{y_1} = - 1$ and
$D = (3, - 1)$ where ${x_2} = 3,{y_2} = - 1$
$\Rightarrow$$CD = \sqrt {{{(3 - ( - 2))}^2} + {{( - 1 - ( - 1))}^2}} = \sqrt {{5^2} + {0^2}} $$ = \sqrt {25} = 5$ Units
Now,
$D = (3, - 1)$where ${x_1} = 3,{y_1} = - 1$ and
$A = (3,4)$where ${x_2} = 3,{y_2} = 4$
$\Rightarrow$$DA = \sqrt {{{(3 - 3)}^2} + {{( - 1 - 4)}^2}} = \sqrt {{0^2} + {{( - 5)}^2}} = \sqrt {25} = 5$ Units
$D$ are $(3, - 1)$
Here we got the side $AB = 5$ unit sq.
Area of the square $ = $ side ×side
Area of the square $ABCD = AB \times AB$
Substituting the side value,
$ \Rightarrow 5 \times 5$
Multiplying both the numbers, we get
Hence,
Area of the square \[ABCD = 25\] Unit sq.
$\therefore $ The area of the square is $25$ unit sq.
Note: Whenever we face such types of problems the key behind the solution lies in the diagrammatic representation of the data provided then awareness about the basic formula can help in getting to the right track to reach the answer.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Why Are Noble Gases NonReactive class 11 chemistry CBSE
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
At which age domestication of animals started A Neolithic class 11 social science CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Write a letter to the principal requesting him to grant class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE