Point { M $}$ Is The Midpoint Of { \overline{AB}$}$. Given That { AM = 3x + 3 $}$ And { AB = 8x - 6 $}$, What Is The Length Of { \overline{AM}$} ? E N T E R Y O U R A N S W E R I N T H E B O X . ?Enter Your Answer In The Box. ? E N T Eryo U R An S W Er In T H E B O X . { \square\$} Units
Finding the Length of {\overline{AM}$}$
Understanding the Problem
In this problem, we are given that point { M $}$ is the midpoint of {\overline{AB}$}$. This means that the distance from point { A $}$ to point { M $}$ is equal to the distance from point { M $}$ to point { B $}$. We are also given the lengths of { AM $}$ and { AB $}$ in terms of the variable { x $}$. Our goal is to find the length of {\overline{AM}$}$.
Recalling the Midpoint Formula
To find the length of {\overline{AM}$}$, we need to recall the midpoint formula. The midpoint formula states that the coordinates of the midpoint of a line segment are the average of the coordinates of the two endpoints. In this case, we are given the lengths of { AM $}$ and { AB $}$, so we can use the midpoint formula to find the length of {\overline{AM}$}$.
Setting Up the Equation
We know that the length of {\overline{AM}$}$ is equal to the length of {\overline{MB}$. We can set up an equation using the given information:
[$ AM = MB $}$
Substituting the given values, we get:
{ 3x + 3 = 8x - 6 $}$
Solving for { x $}$
To solve for { x $}$, we can add { 6 $}$ to both sides of the equation:
{ 3x + 9 = 8x $}$
Subtracting { 3x $}$ from both sides, we get:
{ 9 = 5x $}$
Dividing both sides by { 5 $}$, we get:
{ x = \frac{9}{5} $}$
Finding the Length of {\overline{AM}$}$
Now that we have found the value of { x $}$, we can substitute it into the expression for the length of {\overline{AM}$}$:
{ AM = 3x + 3 $}$
Substituting { x = \frac{9}{5} $}$, we get:
{ AM = 3(\frac{9}{5}) + 3 $}$
Simplifying, we get:
{ AM = \frac{27}{5} + 3 $}$
Adding { 3 $}$ to { \frac{27}{5} $}$, we get:
{ AM = \frac{27}{5} + \frac{15}{5} $}$
Combining the fractions, we get:
{ AM = \frac{42}{5} $}$
Conclusion
In this problem, we used the midpoint formula to find the length of {\overline{AM}$}$. We set up an equation using the given information and solved for { x $}$. Then, we substituted the value of { x $}$ into the expression for the length of {\overline{AM}$}$ and simplified to find the final answer.
The Final Answer
The length of {\overline{AM}$}$ is { \frac{42}{5} $}$ units.
Key Takeaways
- The midpoint formula states that the coordinates of the midpoint of a line segment are the average of the coordinates of the two endpoints.
- To find the length of a line segment, we can use the midpoint formula and set up an equation using the given information.
- Solving for the variable { x $}$ allows us to substitute it into the expression for the length of the line segment and simplify to find the final answer.
Common Mistakes
- Failing to set up the equation correctly
- Not solving for the variable { x $}$ correctly
- Not substituting the value of { x $}$ into the expression for the length of the line segment correctly
Real-World Applications
- The midpoint formula has many real-world applications, such as finding the midpoint of a line segment in a coordinate plane.
- The concept of finding the length of a line segment is used in many fields, such as engineering, architecture, and physics.
Practice Problems
- Find the length of {\overline{BC}$] given that [$ B $}$ is the midpoint of {\overline{AC}$] and [$ AC = 12x - 8 $}$.
- Find the length of {\overline{DE}$] given that [$ D $}$ is the midpoint of {\overline{EF}$] and [$ EF = 9x + 6 $}$.
Conclusion
In this article, we used the midpoint formula to find the length of {\overlineAM}$}$. We set up an equation using the given information and solved for { x $}$. Then, we substituted the value of { x $}$ into the expression for the length of {\overline{AM}$}$ and simplified to find the final answer. We also discussed the key takeaways, common mistakes, and real-world applications of the midpoint formula.
**Q&A$}$**
Frequently Asked Questions
In this article, we will answer some of the most frequently asked questions about finding the length of {\overline{AM}$}$.
Q: What is the midpoint formula?
A: The midpoint formula is a mathematical formula that states that the coordinates of the midpoint of a line segment are the average of the coordinates of the two endpoints.
Q: How do I find the length of {\overline{AM}$}$ using the midpoint formula?
A: To find the length of {\overline{AM}$}$ using the midpoint formula, you need to set up an equation using the given information and solve for the variable { x $}$. Then, you can substitute the value of { x $}$ into the expression for the length of {\overline{AM}$}$ and simplify to find the final answer.
Q: What if I don't know the value of { x $}$?
A: If you don't know the value of { x $}$, you can use the given information to set up an equation and solve for { x $}$. Then, you can substitute the value of { x $}$ into the expression for the length of {\overline{AM}$}$ and simplify to find the final answer.
Q: Can I use the midpoint formula to find the length of any line segment?
A: Yes, you can use the midpoint formula to find the length of any line segment. However, you need to make sure that the line segment has a midpoint and that the coordinates of the two endpoints are known.
Q: What if the line segment is not a straight line?
A: If the line segment is not a straight line, you cannot use the midpoint formula to find its length. In this case, you need to use a different method, such as the Pythagorean theorem, to find the length of the line segment.
Q: Can I use the midpoint formula to find the length of a line segment in a 3D space?
A: Yes, you can use the midpoint formula to find the length of a line segment in a 3D space. However, you need to make sure that the coordinates of the two endpoints are known and that the line segment has a midpoint.
Q: What if I make a mistake in my calculations?
A: If you make a mistake in your calculations, you can try to identify the error and correct it. If you are still having trouble, you can ask for help from a teacher or a tutor.
Q: Can I use the midpoint formula to find the length of a line segment in a coordinate plane?
A: Yes, you can use the midpoint formula to find the length of a line segment in a coordinate plane. However, you need to make sure that the coordinates of the two endpoints are known and that the line segment has a midpoint.
Q: What if I don't have a calculator?
A: If you don't have a calculator, you can try to simplify the expression for the length of the line segment and then use algebraic manipulations to find the final answer.
Q: Can I use the midpoint formula to find the length of a line segment in a real-world application?
A: Yes, you can use the midpoint formula to find the length of a line segment in a real-world application. For example, you can use it to find the length of a road or a bridge.
Conclusion
In this article, we answered some of the most frequently asked questions about finding the length of {\overline{AM}$}$. We discussed the midpoint formula, how to find the length of a line segment using the midpoint formula, and some common mistakes to avoid. We also provided some examples and practice problems to help you understand the concept better.
Practice Problems
- Find the length of {\overline{BC}$] given that [$ B $}$ is the midpoint of {\overline{AC}$] and [$ AC = 12x - 8 $}$.
- Find the length of {\overline{DE}$] given that [$ D $}$ is the midpoint of {\overline{EF}$] and [$ EF = 9x + 6 $}$.
Real-World Applications
- Finding the length of a road or a bridge
- Finding the length of a line segment in a 3D space
- Finding the length of a line segment in a coordinate plane
Common Mistakes
- Failing to set up the equation correctly
- Not solving for the variable { x $}$ correctly
- Not substituting the value of { x $}$ into the expression for the length of the line segment correctly
Conclusion
In this article, we provided some answers to frequently asked questions about finding the length of {\overline{AM}$}$. We discussed the midpoint formula, how to find the length of a line segment using the midpoint formula, and some common mistakes to avoid. We also provided some examples and practice problems to help you understand the concept better.