How do you write the equation of the perpendicular bisector of the segment with the given endpoints #(2,5)# and #(4,9)#?

Question

Savannah Aguilar · Accepted Answer

#x+2y-17=0#.

Let #P(x,y)# be any arbitrary point on the #bot#-bisector of segment #AB#, where, #A(2,5) and B(4,9).# Clearly, dist.#PA#=dist.#PB# #:. PA^2=PB^2# #:. (x-2)^2+(y-5)^2=(x-4)^2+(y-9)^2# #:. -4x+4-10y+25=-8x+16-18y+81# #:. 4x+8y-68=0, or, x+2y-17=0# Given that this relationship is valid for any point on the #bot-#bisector of the seg, it is the reqd. en. of the #bot-#bisector.

Autumn Armstrong · Answer

The equation is #y=-(1/2)x +17/2#

To do that, you must determine the slope of the segment, the middle point (M), the slope of the perpendicular, and, lastly, the equation of the line that passes through M and has a slope perpendicular to the segment.

Middle Point M #x_M=(x_A+x_B)/2=(2+4)/2=3# #y_M=(y_A+y+B)/2=(5+6)/2=7# => M (3,7)

Segment's slope #k=(Delta y)/(Delta x)=(y_B-y_A)/(x_B-x_A)=(9-5)/(4-2)=2#

Perpendicular line's slope #p=-1/k=-1/2#

Equation of perpendicular line passing through point M #y-y_M=p(x-x_M)# #y-7=(-1/2)(x-3)# #y=(-1/2)x+3/2+7# #y=-(1/2)x+17/2#

Violet Aldrich · Answer

undefined To write the equation of the perpendicular bisector of the segment with endpoints (2,5) and (4,9), follow these steps:

1. Find the midpoint of the segment by averaging the x-coordinates and y-coordinates of the endpoints.
2. Calculate the slope of the segment connecting the given endpoints.
3. Determine the negative reciprocal of the slope found in step 2, which will be the slope of the perpendicular bisector.
4. Use the midpoint found in step 1 and the slope from step 3 to write the equation of the perpendicular bisector in point-slope form.
5. Convert the equation to the desired form, such as slope-intercept form or standard form, if necessary.

By following these steps, you can write the equation of the perpendicular bisector of the segment with the given endpoints.