A line segment is bisected by a line with the equation # - 3 y + 5 x = 8 #. If one end of the line segment is at #( 7 , 9 )#, where is the other end?
Coordinates of the other end point
The bisecting line is assumed to be a perpendicular bisector.
Equations (1) and (2) solved,
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To find the other end of the line segment bisected by the line ( -3y + 5x = 8 ), knowing that one end is at (7, 9), follow these steps:
- Find the slope of the line ( -3y + 5x = 8 ) by rearranging it into slope-intercept form ( y = mx + b ), where ( m ) is the slope.
- Use the fact that the line bisects the line segment to find the midpoint of the line segment.
- Use the midpoint formula to find the coordinates of the other end of the line segment.
Let's proceed with these steps:
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Rearrange the equation ( -3y + 5x = 8 ) into slope-intercept form: [ -3y = -5x + 8 ] [ y = \frac{5}{3}x - \frac{8}{3} ]
The slope of the line is ( m = \frac{5}{3} ).
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Since the line bisects the line segment, its slope is perpendicular to the segment. Thus, the negative reciprocal of ( \frac{5}{3} ) is the slope of the line segment.
The negative reciprocal of ( \frac{5}{3} ) is ( -\frac{3}{5} ).
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Use the midpoint formula to find the other end of the line segment: [ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]
Given one end at (7, 9) and the midpoint, we can find the other end.
Let ( (x, y) ) be the coordinates of the other end. [ \text{Midpoint} = \left( \frac{7 + x}{2}, \frac{9 + y}{2} \right) ]
Since the line bisects the line segment, its midpoint must lie on the line ( -3y + 5x = 8 ). Plug in the coordinates of the midpoint into the equation of the line and solve for ( x ) and ( y ).
[ -3\left(\frac{9 + y}{2}\right) + 5\left(\frac{7 + x}{2}\right) = 8 ]
Simplify and solve for ( x ) and ( y ).
After finding the values of ( x ) and ( y ), you will have the coordinates of the other end of the line segment.
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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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