Question

Find the distance between the points
(i) C ($-$1,$-$2), D(4,7)
(ii) E ($-$6, 13), F (9, $-$7)
(iii) G (m+n, m$-$n), H(m$-$n, m+n)

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\text{The}\mathrm{given}\mathrm{points}\text{are}\mathrm{C}\left(-1,-2\right)\mathrm{and}\mathrm{D}\left(4,7\right). \\ \text{From}\mathrm{these}\mathrm{points}, \\ {x}_1=-1,y_1=-2,x_2=4,y_2=7 \\ \mathrm{By}\mathrm{distance}\mathrm{formula}, \end{gathered}$$

$$\begin{gathered} \mathrm{CD}=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2} \\ =\sqrt{{\left[4-\left(-1\right)\right]}^2+{\left[7-\left(-2\right)\right]}^2} \\ =\sqrt{{\left(5\right)}^2+{\left(9\right)}^2} \\ =\sqrt{25+81} \\ =\sqrt{106} \\ \therefore \mathrm{CD}=\sqrt{106} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\text{The}\mathrm{given}\mathrm{points}\text{are}\mathrm{E}\left(-6,13\right)\mathrm{and}\mathrm{F}\left(9,-7\right). \\ \text{From}\mathrm{these}\mathrm{points}, \\ {x}_1=-6,y_1=13,x_2=9,y_2=-7. \\ \mathrm{By}\mathrm{distance}\mathrm{formula}, \end{gathered}$$

$$\begin{gathered} \mathrm{EF}=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2} \\ =\sqrt{{\left[9-\left(-6\right)\right]}^2+{\left(-7-13\right)}^2} \\ =\sqrt{{\left(15\right)}^2+{\left(-20\right)}^2} \\ =\sqrt{225+400} \\ =\sqrt{625} \\ =25 \\ \therefore \mathrm{EF}=25 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)\text{The}\mathrm{given}\mathrm{points}\text{are}\mathrm{G}\left(m+n,m-n\right)\mathrm{and}\mathrm{H}\left(m-n,m+n\right). \\ \text{From}\mathrm{these}\mathrm{points}, \\ {x}_1=m+n,y_1=m-n,x_2=m-n,y_2=m+n. \\ \mathrm{By}\mathrm{distance}\mathrm{formula}, \end{gathered}$$

$$\begin{gathered} \mathrm{GH}=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2} \\ =\sqrt{{\left[\left(m-n\right)-\left(m+n\right)\right]}^2+{\left[\left(m+n\right)-\left(m-n\right)\right]}^2} \\ =\sqrt{{\left(-2n\right)}^2+{\left(2n\right)}^2} \\ =\sqrt{4n^2+4n^2} \\ =2\sqrt{2}n \\ \therefore \mathrm{GH}=2\sqrt{2}n \end{gathered}$$