Let A be #(−3,5)# and B be #(5,−10))#. Find: (1) the length of segment #bar(AB)# (2) the midpoint #P# of #bar(AB)# (3) the point #Q# which splits #bar(AB)# in the ratio #2:5#?

1 Answer
Shwetank Mauria
Mar 1, 2017

(1) the length of the segment #bar(AB)# is #17#
(2) Midpoint of #bar(AB)# is #(1,-7 1/2)#
(3) The coordinates of the point #Q# which splits #bar(AB)# in the ratio #2:5# are #(-5/7,5/7)#

Explanation:

If we have two points #A(x_1,y_1)# and #B(x_2,y_2)#, length of #bar(AB)# i.e. distance between them is given by

#sqrt((x_2-x_1)^2+(x_2-x_1)^2)#

and coordinates of the point #P# that divides the segment #bar(AB)# joining these two points in the ratio #l:m# are

#((lx_2+mx_1)/(l+m),(lx_2+mx_1)/(l+m))#

and as midpoint divided segment in ratio #1:1#, its coordinated would be #((x_2+x_1)/2,(x_2+x_1)/2)#

As we have #A(-3,5)# and #B(5,-10)#

(1) the length of the segment #bar(AB)# is

#sqrt((5-(-3))^2+((-10)-5)^2)#

= #sqrt(8^2+(-15)^2)=sqrt(65+225)=sqrt289=17#

(2) Midpoint of #bar(AB)# is #((5-3)/2,(-10-5)/2)# or #(1,-7 1/2)#

(3) The coordinates of the point #Q# which splits #bar(AB)# in the ratio #2:5# are

#((2xx5+5xx(-3))/7,(2xx(-10)+5xx5)/7)# or #((10-15)/7,(-20+25)/7)#

i.e. #(-5/7,5/7)#

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