Answer:
- [tex]\mathrm{Distance\:between\:}\left(-3,\:10\right)\mathrm{\:and\:}\left(7,\:15\right)=\:5\sqrt{5}[/tex]
- [tex]\mathrm{Midpoint\:of}\left(-3,\:10\right)\mathrm{\:and\:}\left(7,\:15\right)=\:\left(2,\:\frac{25}{2}\right)[/tex]
Step-by-step explanation:
Given the two points
Determining the distance between the points
[tex]\mathrm{Compute\:the\:distance\:between\:}\left(x_1,\:y_1\right),\:\left(x_2,\:y_2\right):\quad \sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}[/tex]
[tex]\mathrm{The\:distance\:between\:}\left(-3,\:10\right)\mathrm{\:and\:}\left(7,\:15\right)\mathrm{\:is\:}[/tex]
[tex]=\sqrt{\left(7-\left(-3\right)\right)^2+\left(15-10\right)^2}[/tex]
[tex]=\sqrt{\left(7+3\right)^2+\left(15-10\right)^2}[/tex]
[tex]=\sqrt{10^2+5^2}[/tex]
[tex]=\sqrt{100+25}[/tex]
[tex]=\sqrt{125}[/tex]
[tex]=\sqrt{5^3}[/tex]
[tex]=\sqrt{5^2\cdot \:5}[/tex]
[tex]=5\sqrt{5}[/tex]
Therefore,
[tex]\mathrm{Distance\:between\:}\left(-3,\:10\right)\mathrm{\:and\:}\left(7,\:15\right)=\:5\sqrt{5}[/tex]
Determining the midpoints between the points
Given the two points
[tex]\mathrm{Midpoint\:of\:}\left(x_1,\:y_1\right),\:\left(x_2,\:y_2\right):\quad \left(\frac{x_2+x_1}{2},\:\:\frac{y_2+y_1}{2}\right)[/tex]
[tex]\left(x_1,\:y_1\right)=\left(-3,\:10\right),\:\left(x_2,\:y_2\right)=\left(7,\:15\right)[/tex]
[tex]=\left(\frac{7-3}{2},\:\frac{15+10}{2}\right)[/tex]
[tex]=\left(2,\:\frac{25}{2}\right)[/tex]
Therefore,
[tex]\mathrm{Midpoint\:of}\left(-3,\:10\right)\mathrm{\:and\:}\left(7,\:15\right)=\:\left(2,\:\frac{25}{2}\right)[/tex]
