Question 446070
Given two points (x1,y1), (x2,y2),
Midpoint formula: (x,y) of midpoint = ({{{(x1+x2)/2}}}, {{{(y1+y2)/2}}})
Distance formula: distance = {{{sqrt((x2-x1)^2 + (y2-y1)^2)}}}
***
1a. (midpoint)
(x,y) = ({{{(x1+x2)/2}}}, {{{(y1+y2)/2}}})
(x,y) = ({{{(4+6)/2}}}, {{{(2+10)/2}}})
(x,y) = ({{{10/2}}}, {{{12/2}}})
(x,y) = (5, 6)
Therefore, the midpoint between (4, 2) and (6, 10) is (5, 6).
***
1b. (distance)
{{{sqrt((x2-x1)^2 + (y2-y1)^2)}}}
{{{sqrt((6-4)^2 + (10-2)^2)}}}
{{{sqrt((2)^2 + (8)^2)}}}
{{{sqrt(4 + 64)}}}
{{{sqrt(68)}}}
Factor: {{{sqrt(2*2*17)}}}
You can pull out one number for every pair within a square root’s factors. Since there are two 2s, you can pull out a two: {{{2(sqrt(17))}}}
Therefore, the distance between (4, 2) and (6, 10) is {{{2sqrt(17)}}}
***
2a. (midpoint)
(x,y) = ({{{(x1+x2)/2}}}, {{{(y1+y2)/2}}})
(x,y) = ({{{(-3+4)/2}}}, {{{(5+9)/2}}})
(x,y) = ({{{1/2}}}, {{{14/2}}})
(x,y) = ({{{1/2}}}, 7)
Therefore, the midpoint between (-3, 5) and (4, 9) is ({{{1/2}}}, 7).
***
2b. (distance)
{{{sqrt((x2-x1)^2 + (y2-y1)^2)}}}
{{{sqrt((4-(-3))^2 + (9-5)^2)}}}
{{{sqrt((4+3)^2 + (4)^2)}}}
{{{sqrt((7)^2 + 16)}}}
{{{sqrt(49 + 16)}}}
{{{sqrt(65)}}}
Factor: {{{sqrt(5*13)}}}
There are no pairs of numbers so you cannot simplify.
Therefore, the distance between (-3, 5) and (4, 9) is {{{sqrt(65)}}}.