Question

Find the other endpoint of the line segment with the given endpoint and midpoint. 3) Endpoint: (-5, 4), midpoint:(-10, -6) 4) Endpoint: (-8, 8), midpoint: (5.-3)

Answer 1

We have to find the other endpoint of the line segment with:

`\begin{gathered} \text{Endpoint}=(-5,4) \\ \text{Midpoint}=(-10,-6) \end{gathered}`

For doing so, we will have to find the x and y-coordinates of the point.

First, we will find the line for which the two points pass through. Then, we will find the x-coordinate of the endpoint, using the definition of midpoint, and finally, we will use the x-coordinate and the function for finding the y-coordinate of the point.

Finding the line

We will find the slope, and the y-intercept. For the slope, we use the formula:

`m=\frac{y_2-y_1_{}}{x_2-x_1_{}}`

where (x₁,y₁) and (x₂,y₂) are the coordinates of the points. Using it we find:

`m=(4-(-6))/(-5-(-10))=(10)/(5)=2`

This means that the slope is 2. Now, for the y-intercept, we replace on the general slope-intercept formula:

`y=mx+b`

And we get that:

`\begin{gathered} 4=2(-5)+b \\ 4=-10+b \\ 4+10=b \\ 14=b \end{gathered}`

We get that the equation of the line is:

`y=2x+14`

Finding the x-coordinate

For this step, we know that the distance in the x-coordinates between the midpoint and the endpoint is:

`\lvert-10-(-5)\rvert=\lvert-10+5\rvert=\lvert-5\rvert=5`

This means that the distance in the x-coordinates between the midpoint and the other endpoint is also 5. Thus, as it is moving to the left, the x-coordinate will be -10-5=-15.

Finding the y-coordinate

For this step, we replace the x-coordinate onto the function obtained on the step 1, and we will get the y-coordinate of the other endpoint.