Question

Find the distance between each of the following pairs of points.
$\sf \: 1) \: A ( 2 , 3 ) \: , B ( 4 , 1 )$

$\sf \: 2) \: P( -5 , 7 ) \: , Q( -1 , 3)$

$\sf \: 3) \: T( -3 \: , 6 ) \: , R( 9 \: , -10)$
​

Answer 1

Step-by-step explanation:

According to the Question .

We have to calculate the distance between the given pairs of points .

Using Distance Formula

$\\ \star{\boxed{\purple{\bf{Distance \: = \sqrt{(x_{2} - x_{1}) {}^{2} + (y_{2} - y_{1}) {}^{2} } }}}}$

$\sf \: 1) \: A ( 2 , 3 ) \: , B ( 4 , 1 )$

$\sf \implies \: AB \: = \sqrt{(4 - 2) {}^{2} + (1 - 3)} {}^{2} \\ \\ \sf \implies \: AB \: = \sqrt{( 2) {}^{2} + (-2) {}^{2} } \\ \\ \sf \implies \: AB \: = \sqrt{ 4 + 4} \\ \\ \sf \implies \:AB \: = \sqrt{8} \\ \\ \sf \implies \: AB \: = 2 \sqrt{2}$

$\sf \: 2) \: P( -5 , 7 ) \: , Q( -1 , 3)$

On substituting the value we get

$\sf \implies \: PQ \: = \sqrt{ ({ - 1 + 5})^{2} + ({3 - 7})^{2} } \\ \\ \sf \implies \: PQ \: = \sqrt{ {(4)}^{2} + {( - 4)}^{2} } \\ \\ \sf \implies \: PQ \: = \sqrt{16 + 16} \\ \\ \sf \implies \: PQ \: = 4\sqrt{2}$

$\sf \: 3) \: T( -3 \: , 6 ) \: , R( 9 \: , -10)$

On substituting the value we get

$\sf \implies \: TR = \sqrt{ {(9 + 3)}^{2} {( - 10 - 6)}^{2} } \\ \\ \sf \implies \: TR \: = \sqrt{ {(12)}^{2} + { ( - 16)}^{2} } \\ \\ \sf \implies \: TR \: = \sqrt{144 + 256} \\ \\ \sf \implies \: TR \: = \sqrt{400} \\ \\ \sf \implies \: TR \: = 20$

Answer 2

$\huge \rm \underline{Solution - 1}$

$\\ \star{\boxed{\purple{\bf{Assumption}}}}$

$\rm Consider \: W \: ( x_1 ,y_1) \: and \: X \: ( x_2 ,y_2) \: be \: the \: given \: points \:$

$\rm \therefore \: x_1 = 2, \: y_1 = 3 \: and \: x_2 = 4 , \: y_2 = 1$

$\\ \star{\boxed{\purple{\bf{Now}}}}$

By using the distance formula ,

$\rm d(W , X) = \sqrt{( x_2 - y_1 {)}^{2} \: + \: {( y_2 - y_1 {)}^{2} \: } }$

➡ Substitute the given values in above formula and solve

$\rm \implies \sqrt{(4 -2 {)}^{2} \: + \: {( 1 - 3{)}^{2} \: } }$

$\rm \implies \sqrt{(2 {)}^{2} \: + \: {( - 2{)}^{2} \: } }$

$\rm \implies \sqrt{4 + 4} = \sqrt{8}$

$\rm \therefore \: d(W , X) = 2 \sqrt{2} \: units$

$\\ \star{\boxed{\purple{\bf{Henceforth}}}}$

➡ The required distance between the points W and X is 2 √2 units

$\huge \rm \underline{Solution - 2}$

$\\ \star{\boxed{\purple{\bf{Assumption}}}}$

$\rm Consider \: M \: ( x_1 ,y_1) \: and \: N \: ( x_2 ,y_2) \: be \: the \: given \: points \:$

$\rm \therefore \: x_1 = - 5, \: y_1 = 7\: and \: x_2 = - 1, \: y_2 = 3$

$\\ \star{\boxed{\purple{\bf{Now}}}}$

By using the distance formula ,

$\rm d(M, N) = \sqrt{( x_2 - y_1 {)}^{2} \: + \: {( y_2 - y_1 {)}^{2} \: } }$

➡ Substitute the given values in above formula and solve

$\rm \implies \sqrt{ [- 1 - ( - {5}) {]}^{2} \ \: + \: {( 3- 7{)}^{2} \: } }$

$\rm \implies \sqrt{ (- 1 + {5}{)}^{2} \ \: + \: {( 3- 7{)}^{2} \: } }$

$\rm \implies \sqrt{ {4}^{2} \ \: + \: {( - 4{)}^{2} \: } }$

$\rm \implies \sqrt{16 + 16 }$

$\rm \implies \sqrt{32}$

$\rm \therefore \: d(M , N) = 4 \sqrt{2} \: units$

$\\ \star{\boxed{\purple{\bf{Henceforth}}}}$

➡ The required distance between the points W and X is 4 √2 units

$\huge \rm \underline{Solution - 3}$

$\\ \star{\boxed{\purple{\bf{Assumption}}}}$

$\rm Consider \: P \: ( x_1 ,y_1) \: and \: Q\: ( x_2 ,y_2) \: be \: the \: given \: points \:$

$\rm \therefore \: x_1 = - 3, \: y_1 = 6 \: and \: x_2 = 9 , \: y_2 = - 10$

$\\ \star{\boxed{\purple{\bf{Now}}}}$

By using the distance formula ,

$\rm d(P, Q) = \sqrt{( - y_1 {)}^{2} \: + \: {( y_2 - y_1 {)}^{2} \: } }$

➡ Substitute the given values in above formula and solve

$\rm \implies \sqrt{ [9- ( - {3}) {]}^{2} \ \: + \: {( - 10 - 6{)}^{2} \: } }$

$\rm \implies (9 + 3 {)}^{2} + ( - 10 - 6 {)}^{2}$

$\rm \implies {12}^{2} + ( - 16 {)}^{2}$

$\rm \implies \sqrt{144 + 256}$

$\rm \implies \sqrt{400}$

$\rm \therefore \: d(P , Q) = 20 \: units$

$\\ \star{\boxed{\purple{\bf{Henceforth}}}}$

➡ The required distance between the points P and Q is 20 units