Determining The Coordinates Of An EndpointThe Midpoint Of $\overline{PQ}$ Is $M\left(-\frac{1}{2},-1\right$\]. One Endpoint Is $Q(3,-5$\]. Which Equations Calculate The Coordinates Of $P$? Check All That Apply.A.

Introduction

In geometry, the midpoint of a line segment is the point that divides the segment into two equal parts. Given the midpoint and one endpoint of a line segment, we can use the midpoint formula to find the coordinates of the other endpoint. In this article, we will explore how to determine the coordinates of an endpoint using the midpoint formula.

Midpoint Formula

The midpoint formula is given by:

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two endpoints of the line segment.

Given Information

We are given the midpoint of $\overline{PQ}$ as $M\left(-\frac{1}{2},-1\right)$ and one endpoint as $Q(3,-5)$. We need to find the equations that calculate the coordinates of $P$.

Step 1: Write the Midpoint Formula

Using the midpoint formula, we can write:

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)=\left(-\frac{1}{2},-1\right)$$

where $(x_1, y_1)$ are the coordinates of $P$ and $(x_2, y_2)$ are the coordinates of $Q$.

Step 2: Substitute the Given Values

Substituting the given values, we get:

$$\left(\frac{x_1+3}{2},\frac{y_1-5}{2}\right)=\left(-\frac{1}{2},-1\right)$$

Step 3: Equate the x-coordinates

Equate the x-coordinates:

$$\frac{x_1+3}{2}=-\frac{1}{2}$$

Solving for $x_1$, we get:

$$x_1+3=-1$$

$$x_1=-4$$

Step 4: Equate the y-coordinates

Equate the y-coordinates:

$$\frac{y_1-5}{2}=-1$$

Solving for $y_1$, we get:

$$y_1-5=-2$$

$$y_1=-3$$

Conclusion

Therefore, the coordinates of $P$ are $(-4,-3)$.

Answer Options

The equations that calculate the coordinates of $P$ are:

• $x_1=-4$
• $y_1=-3$

These equations are based on the midpoint formula and the given information.

Final Answer

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Introduction

In the previous article, we explored how to determine the coordinates of an endpoint using the midpoint formula. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the midpoint formula?

A: The midpoint formula is given by:

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two endpoints of the line segment.

Q: How do I use the midpoint formula to find the coordinates of an endpoint?

A: To use the midpoint formula, you need to know the coordinates of the midpoint and one endpoint of the line segment. You can then substitute the given values into the midpoint formula and solve for the unknown coordinates.

Q: What if I have two endpoints and I want to find the midpoint?

A: If you have two endpoints and you want to find the midpoint, you can use the midpoint formula in reverse. Simply substitute the coordinates of the two endpoints into the midpoint formula and solve for the midpoint coordinates.

Q: Can I use the midpoint formula to find the distance between two points?

A: No, the midpoint formula is used to find the coordinates of a point that divides a line segment into two equal parts. It is not used to find the distance between two points. To find the distance between two points, you can use the distance formula:

$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

Q: What if I have a line segment with a midpoint and two endpoints, but I want to find the equation of the line that passes through the endpoints?

A: To find the equation of the line that passes through the endpoints, you can use the two-point form of a line:

$$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two endpoints.

Q: Can I use the midpoint formula to find the equation of a circle that passes through the midpoint and two endpoints?

A: No, the midpoint formula is used to find the coordinates of a point that divides a line segment into two equal parts. It is not used to find the equation of a circle. To find the equation of a circle that passes through the midpoint and two endpoints, you can use the general form of a circle:

$$(x-h)^2+(y-k)^2=r^2$$

where $(h, k)$ is the center of the circle and $r$ is the radius.

Conclusion

In this article, we answered some frequently asked questions related to determining the coordinates of an endpoint using the midpoint formula. We hope this article has been helpful in clarifying any doubts you may have had.

Final Answer

The final answer is: $\boxed{(-4,-3)}$