A line segment is bisected by a line with the equation # 4 y - 6 x = 8 #. If one end of the line segment is at #( 8 , 3 )#, where is the other end?

Answer 1

Exact value: #(-28/13,127/13)#

Decimal value (-2.16 9.77)

Bisectors are perpendicular - they bisect the line at a right angle. This means that the gradient of the line segment is the reciprocal of the other line: #4y=6x+8# #y=3/2x+2# has a gradient of #(3/2)# #:.# unknown segment has a gradient of #color(blue)(-2/3)#
So our line segment has a point at #color(red)(8),color(green)(3)# and gradient #color(blue)(-2/3)# #y-y_1=m(x-x_1#) #:. y-color(green)(3)=color(blue)(-2/3)(x-color(red)8)# #y-color(green)(3)=color(blue)(-2/3)x+color(red)16/3# #y=color(blue)(-2/3)x+25/3# is the equation of our line segment.

Currently, locate the other endpoint:

The lines intersect where one equation = the other #:.# where #3/2x+2=-2/3x+25/3# #13/6x=19/3# #x=38/13 # or #~~ 2.92#
Distance between point given #(x=8)# and midpoint found #(x=38/13)# is #66/13# or 5.08 units. So other endpoint of line is at #8-2*(66)/(13)=(-28)/(13)#, #=> y=(127/13)#
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Answer 2

To find the other end of the line segment bisected by the line with the equation (4y - 6x = 8) given one end at (8, 3), follow these steps:

  1. Calculate the slope of the line given by the equation (4y - 6x = 8).
  2. Determine the midpoint of the line segment using the given point and the slope of the line.
  3. Use the midpoint and the given point to find the other end of the line segment.

First, rearrange the equation (4y - 6x = 8) into slope-intercept form (y = mx + b), where (m) is the slope and (b) is the y-intercept.

[4y - 6x = 8] [4y = 6x + 8] [y = \frac{3}{2}x + 2]

The slope of the line is (\frac{3}{2}).

The midpoint formula for a line segment with endpoints ((x_1, y_1)) and ((x_2, y_2)) is given by:

[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]

Given one end at (8, 3), we can use the midpoint formula to find the midpoint.

[ \text{Midpoint} = \left( \frac{8 + x_2}{2}, \frac{3 + y_2}{2} \right) ]

Since the line bisects the segment, the slope of the line from (8, 3) to the midpoint is the negative reciprocal of (\frac{3}{2}), which is (-\frac{2}{3}).

Now, using the point-slope form of a line ((y - y_1 = m(x - x_1))), we can find the equation of the line passing through the given point (8, 3) with the slope (-\frac{2}{3}).

[y - 3 = -\frac{2}{3}(x - 8)]

[y - 3 = -\frac{2}{3}x + \frac{16}{3}]

[y = -\frac{2}{3}x + \frac{16}{3} + 3]

[y = -\frac{2}{3}x + \frac{25}{3}]

Next, set this equation equal to the equation of the line given in the question to find the intersection point.

[ -\frac{2}{3}x + \frac{25}{3} = \frac{3}{2}x + 2 ]

Solve for (x), then substitute the value of (x) back into either equation to find (y). This will give the coordinates of the other end of the line segment.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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