# A line segment is bisected by a line with the equation # - y + 4 x = 3 #. If one end of the line segment is at #( 2 , 6 )#, where is the other end?

Any point on the line

Consider the vertical line

Clearly

and it intersects

Therefore

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Furthermore (as seen in the image below and considering similar triangles)

any point on a line parallel to

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

If the line through

it will have a y-intercept at

and therefore a slope-intercept form of

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The other end of the line segment bisected by the line with the equation (-y + 4x = 3) can be found by using the midpoint formula. First, we need to determine the coordinates of the midpoint, which lies on the given line.

The midpoint formula is given by: [ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]

Given that one end of the line segment is at (2, 6), we can use the midpoint formula to find the midpoint.

Substitute the given point (2, 6) into the equation (-y + 4x = 3) to find the y-coordinate of the midpoint: [ -y + 4(2) = 3 ] [ -y + 8 = 3 ] [ -y = -5 ] [ y = 5 ]

So, the midpoint is at (2, 5). Now, we can use this midpoint and the given end point (2, 6) to find the other end of the line segment.

Let the coordinates of the other end be (a, b). Using the midpoint formula, we have: [ \frac{2 + a}{2} = 2 ] [ 2 + a = 4 ] [ a = 2 ]

[ \frac{6 + b}{2} = 5 ] [ 6 + b = 10 ] [ b = 4 ]

Therefore, the other end of the line segment is at (2, 4).

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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.

- A triangle has corners A, B, and C located at #(4 ,5 )#, #(3 ,6 )#, and #(2 ,9 )#, respectively. What are the endpoints and length of the altitude going through corner C?
- A line segment is bisected by a line with the equation # - 3 y + 5 x = 2 #. If one end of the line segment is at #( 7 , 9 )#, where is the other end?
- A triangle has corners A, B, and C located at #(1 ,3 )#, #(7 ,4 )#, and #(5 , 8 )#, respectively. What are the endpoints and length of the altitude going through corner C?
- What is the orthocenter of a triangle with corners at #(2 ,3 )#, #(9 ,1 )#, and (6 ,3 )#?
- Given point A #(-2,1)# and point B #(1,3)#, how do you find the equation of the line perpendicular to the line AB at its midpoint?

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