Question 1161262

A line in the ({{{x}}},{{{y}}}) coordinate plane contains the points 

{{{P}}}({{{x}}},{{{7}}}) and 
{{{Q}}}({{{1}}},{{{1}}})

a) 
Given that the distance between the points{{{ P}}} and{{{ Q }}}is {{{10 }}}units, what is the value of {{{x}}}?

use distance formula:

{{{d=sqrt((x-x[1])^2+(y-y[1])^2)}}}....plug in given points and distance

{{{10=sqrt((x-1)^2+(7-1)^2)}}}
{{{10=sqrt(x^2-2x+1+6^2)}}}
{{{10^2=x^2-2x+1+36}}}
{{{100=x^2-2x+37}}}
{{{0=x^2-2x+37-100}}}
{{{x^2-2x-63=0}}}
{{{(x - 9) (x + 7) = 0}}}
=> {{{x=9}}} or {{{x=-7}}}

so, {{{P}}}({{{9}}},{{{7}}})  or {{{P}}}({{{-7}}},{{{7}}})


b)

 Use your answer in (a) to find the midpoint of {{{PQ}}}.

if {{{P}}}({{{9}}},{{{7}}}) and 
{{{Q}}}({{{1}}},{{{1}}})

 the midpoint of{{{PQ}}} is 
{{{M}}}=({{{(9+1)/2}}},{{{(7+1)/2}}})=({{{5}}},{{{4}}})

or
if {{{P}}}({{{-7}}},{{{7}}}) and 
{{{Q}}}({{{1}}},{{{1}}})

{{{M}}}=({{{(-7+1)/2}}},{{{(7+1)/2}}})=({{{-3}}},{{{4}}})