# A line segment is bisected by a line with the equation # 4 y - 3 x = 2 #. If one end of the line segment is at #( 7 , 5 )#, where is the other end?

The other point is on the line

(any point on this line will satisfy the given requirement)

Given

and a point

Consider the vertical line through

This vertical line will intersect

where

The distance from

is

So a point

Note that any point on a line parallel to

If

#color(red)("L1")# parallel to#color(green)("L2")#

then#triangle ABC ~=triangle ADE#

#rarr abs(AB):abs(AD)=abs(AC):abs(AE)#

Since

Our required line will also have a slope of

and since it passes through

using the slope-point form, we have

or

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The other end-pt. lies on the line given by the eqn.

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To find the other end of the line segment bisected by the line (4y - 3x = 2), given that one end is at (7, 5), we first need to find the midpoint of the line segment. Then, we can use the midpoint formula to find the coordinates of the other end.

First, let's find the midpoint of the line segment. We know that the midpoint is the average of the coordinates of the endpoints. So, if (x, y) is the midpoint, we have:

[x_{\text{midpoint}} = \frac{x_1 + x_2}{2}] [y_{\text{midpoint}} = \frac{y_1 + y_2}{2}]

Given that one endpoint is at (7, 5), we can substitute these values into the midpoint formula:

[x_{\text{midpoint}} = \frac{7 + x_2}{2}] [y_{\text{midpoint}} = \frac{5 + y_2}{2}]

Now, we need to find the coordinates of the other endpoint, which will be the solution to the system of equations formed by the line equation and the midpoint formula.

Substituting the equation of the line ((4y - 3x = 2)) into the midpoint formula, we get:

[x_{\text{midpoint}} = \frac{7 + x_2}{2} \implies 3x_2 = 14 - x] [y_{\text{midpoint}} = \frac{5 + y_2}{2} \implies 4y_2 = 10 - y]

Now, we can solve this system of equations to find the values of (x) and (y), which represent 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.

- A line segment is bisected by line with the equation # 6 y - 7 x = 3 #. If one end of the line segment is at #(7 ,2 )#, where is the other end?
- A triangle has corners A, B, and C located at #(2 ,5 )#, #(7 ,4 )#, and #(6 ,1 )#, respectively. What are the endpoints and length of the altitude going through corner C?
- A triangle has corners A, B, and C located at #(4 ,3 )#, #(9 ,5 )#, and #(6 ,2 )#, respectively. What are the endpoints and length of the altitude going through corner C?
- What is the centroid of a triangle with corners at #(2, 7 )#, #(1,5 )#, and #(7 , 5 )#?
- What is the centroid of a triangle with corners at #(9 , 2 )#, #(6 , 4 )#, and #(1 , 3 )#?

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