Segment { AB $}$ Has Point { A $}$ Located At { (4, 2)$}$. If The Distance From { A $}$ To { B $}$ Is 3 Units, Which Of The Following Could Be Used To Calculate The Coordinates For Point [$ B

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Introduction

In mathematics, calculating the coordinates of a point on a segment is a fundamental concept that has numerous applications in various fields, including geometry, trigonometry, and physics. In this article, we will explore how to calculate the coordinates of point { B $}$ given the coordinates of point { A $}$ and the distance between the two points.

Understanding the Problem

We are given that point { A $}$ is located at {(4, 2)$}$ and the distance from { A $}$ to { B $}$ is 3 units. Our goal is to find the possible coordinates for point { B $}$.

The Distance Formula

The distance formula is a fundamental concept in mathematics that calculates the distance between two points in a coordinate plane. The formula is given by:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

where { d $}$ is the distance between the two points, and { (x_1, y_1) $}$ and { (x_2, y_2) $}$ are the coordinates of the two points.

Calculating the Coordinates of Point { B $}$

To calculate the coordinates of point { B $}$, we can use the distance formula and the coordinates of point { A $}$. Let's assume that the coordinates of point { B $}$ are { (x, y) $}$. We can then use the distance formula to set up an equation:

3=(x−4)2+(y−2)23 = \sqrt{(x - 4)^2 + (y - 2)^2}

Squaring both sides of the equation, we get:

9=(x−4)2+(y−2)29 = (x - 4)^2 + (y - 2)^2

Expanding the equation, we get:

9=x2−8x+16+y2−4y+49 = x^2 - 8x + 16 + y^2 - 4y + 4

Rearranging the equation, we get:

x2−8x+y2−4y=−9+20x^2 - 8x + y^2 - 4y = -9 + 20

Simplifying the equation, we get:

x2−8x+y2−4y=11x^2 - 8x + y^2 - 4y = 11

Possible Coordinates for Point { B $}$

To find the possible coordinates for point { B $}$, we need to find the values of { x $}$ and { y $}$ that satisfy the equation. We can do this by using algebraic methods or by graphing the equation.

Algebraic Method

One way to solve the equation is to use algebraic methods. We can start by completing the square for the { x $}$ and { y $}$ terms:

x2−8x+16+y2−4y+4=11+16+4x^2 - 8x + 16 + y^2 - 4y + 4 = 11 + 16 + 4

x2−8x+16=31−y2+4y−4x^2 - 8x + 16 = 31 - y^2 + 4y - 4

(x−4)2=31−(y2−4y+4)(x - 4)^2 = 31 - (y^2 - 4y + 4)

(x−4)2=31−(y−2)2(x - 4)^2 = 31 - (y - 2)^2

Taking the square root of both sides, we get:

x−4=±31−(y−2)2x - 4 = \pm \sqrt{31 - (y - 2)^2}

Solving for { x $}$, we get:

x=4±31−(y−2)2x = 4 \pm \sqrt{31 - (y - 2)^2}

Graphical Method

Another way to solve the equation is to graph the equation. We can graph the equation by plotting the points { (4, 2) $}$ and { (x, y) $}$ and drawing a line through the two points.

Conclusion

In conclusion, we have shown how to calculate the coordinates of point { B $}$ given the coordinates of point { A $}$ and the distance between the two points. We have used the distance formula and algebraic methods to solve the equation and find the possible coordinates for point { B $}$.

Possible Solutions

There are infinitely many possible solutions to the equation. However, we can find some possible solutions by using the algebraic method.

Let's assume that { y = 2 $}$. Then, we can substitute this value into the equation and solve for { x $}$:

x2−8x=11x^2 - 8x = 11

Solving for { x $}$, we get:

x=4±11x = 4 \pm \sqrt{11}

Therefore, one possible solution is { (4 + \sqrt{11}, 2) $}$.

Similarly, let's assume that { y = 6 $}$. Then, we can substitute this value into the equation and solve for { x $}$:

x2−8x=−9x^2 - 8x = -9

Solving for { x $}$, we get:

x=4±25x = 4 \pm \sqrt{25}

Therefore, one possible solution is { (4 + 5, 6) $}$.

Final Answer

In conclusion, the possible coordinates for point { B $}$ are { (4 + \sqrt{11}, 2) $}$ and { (4 + 5, 6) $}$.

References

Additional Resources

Introduction

In our previous article, we explored how to calculate the coordinates of point { B $}$ given the coordinates of point { A $}$ and the distance between the two points. In this article, we will answer some frequently asked questions related to segment { AB $}$ coordinate calculation.

Q&A

Q: What is the distance formula?

A: The distance formula is a fundamental concept in mathematics that calculates the distance between two points in a coordinate plane. The formula is given by:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

where { d $}$ is the distance between the two points, and { (x_1, y_1) $}$ and { (x_2, y_2) $}$ are the coordinates of the two points.

Q: How do I calculate the coordinates of point { B $}$ given the coordinates of point { A $}$ and the distance between the two points?

A: To calculate the coordinates of point { B $}$, you can use the distance formula and algebraic methods. Let's assume that the coordinates of point { B $}$ are { (x, y) $}$. We can then use the distance formula to set up an equation:

3=(x−4)2+(y−2)23 = \sqrt{(x - 4)^2 + (y - 2)^2}

Squaring both sides of the equation, we get:

9=(x−4)2+(y−2)29 = (x - 4)^2 + (y - 2)^2

Expanding the equation, we get:

9=x2−8x+16+y2−4y+49 = x^2 - 8x + 16 + y^2 - 4y + 4

Rearranging the equation, we get:

x2−8x+y2−4y=−9+20x^2 - 8x + y^2 - 4y = -9 + 20

Simplifying the equation, we get:

x2−8x+y2−4y=11x^2 - 8x + y^2 - 4y = 11

Q: What are some possible solutions to the equation?

A: There are infinitely many possible solutions to the equation. However, we can find some possible solutions by using the algebraic method.

Let's assume that { y = 2 $}$. Then, we can substitute this value into the equation and solve for { x $}$:

x2−8x=11x^2 - 8x = 11

Solving for { x $}$, we get:

x=4±11x = 4 \pm \sqrt{11}

Therefore, one possible solution is { (4 + \sqrt{11}, 2) $}$.

Similarly, let's assume that { y = 6 $}$. Then, we can substitute this value into the equation and solve for { x $}$:

x2−8x=−9x^2 - 8x = -9

Solving for { x $}$, we get:

x=4±25x = 4 \pm \sqrt{25}

Therefore, one possible solution is { (4 + 5, 6) $}$.

Q: How do I graph the equation?

A: To graph the equation, you can plot the points { (4, 2) $}$ and { (x, y) $}$ and draw a line through the two points.

Q: What are some real-world applications of segment { AB $}$ coordinate calculation?

A: Segment { AB $}$ coordinate calculation has numerous real-world applications, including:

  • Calculating distances between two points in a coordinate plane
  • Finding the midpoint of a line segment
  • Determining the slope of a line
  • Calculating the area of a triangle

Q: What are some common mistakes to avoid when calculating the coordinates of point { B $}$?

A: Some common mistakes to avoid when calculating the coordinates of point { B $}$ include:

  • Not using the correct distance formula
  • Not squaring both sides of the equation
  • Not expanding the equation correctly
  • Not solving for the correct variables

Conclusion

In conclusion, segment { AB $}$ coordinate calculation is a fundamental concept in mathematics that has numerous real-world applications. By understanding the distance formula and algebraic methods, you can calculate the coordinates of point { B $}$ given the coordinates of point { A $}$ and the distance between the two points. Remember to avoid common mistakes and use the correct methods to ensure accurate results.

References

Additional Resources