If One End Of A Line, Whose Length Is 13 Units, Is The Point \[$(4, 8)\$\] And The Ordinate Of The Other End Is 3, What Is Its Abscissa?

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Understanding the Problem

In this problem, we are given the length of a line and the coordinates of one of its ends. We need to find the coordinates of the other end. The length of the line is 13 units, and the coordinates of one end are (4, 8). The ordinate (y-coordinate) of the other end is 3. We need to find the abscissa (x-coordinate) of the other end.

Using the Distance Formula

To solve this problem, we can use the distance formula, which is derived from the Pythagorean theorem. The distance formula is:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where d is the distance between the two points, and (x1, y1) and (x2, y2) are the coordinates of the two points.

Applying the Distance Formula

In this problem, we know the distance (d = 13) and the coordinates of one end (x1 = 4, y1 = 8). We also know the ordinate (y2 = 3) of the other end. We need to find the abscissa (x2) of the other end. Plugging in the values, we get:

13 = √((x2 - 4)^2 + (3 - 8)^2)

Simplifying the Equation

Simplifying the equation, we get:

13 = √((x2 - 4)^2 + (-5)^2)

Expanding the squared terms, we get:

13 = √(x2^2 - 8x2 + 16 + 25)

Combine like terms:

13 = √(x2^2 - 8x2 + 41)

Squaring Both Sides

Squaring both sides of the equation, we get:

169 = x2^2 - 8x2 + 41

Rearranging the Equation

Rearranging the equation, we get:

x2^2 - 8x2 - 128 = 0

Solving the Quadratic Equation

This is a quadratic equation in x2. We can solve it using the quadratic formula:

x2 = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -8, and c = -128. Plugging in the values, we get:

x2 = (8 ± √((-8)^2 - 4(1)(-128))) / 2(1)

x2 = (8 ± √(64 + 512)) / 2

x2 = (8 ± √576) / 2

x2 = (8 ± 24) / 2

Finding the Two Possible Values of x2

We have two possible values of x2:

x2 = (8 + 24) / 2 = 32 / 2 = 16

x2 = (8 - 24) / 2 = -16 / 2 = -8

Choosing the Correct Value of x2

Since the length of the line is 13 units, the x-coordinate of the other end must be greater than 4. Therefore, the correct value of x2 is 16.

Conclusion

In this problem, we used the distance formula to find the abscissa of the other end of a line, given the length of the line and the coordinates of one end. We solved the quadratic equation to find the two possible values of x2 and chose the correct value based on the given information.

Key Takeaways

  • The distance formula can be used to find the distance between two points in a coordinate plane.
  • The quadratic formula can be used to solve quadratic equations.
  • When solving a quadratic equation, we need to consider both the positive and negative roots.

Real-World Applications

This problem has real-world applications in various fields, such as:

  • Surveying: In surveying, we need to find the coordinates of points on the ground. The distance formula can be used to find the distance between two points.
  • Navigation: In navigation, we need to find the shortest distance between two points. The distance formula can be used to find the shortest distance.
  • Engineering: In engineering, we need to find the coordinates of points in a coordinate plane. The distance formula can be used to find the distance between two points.

Final Answer

Q: What is the distance formula, and how is it used to find the distance between two points?

A: The distance formula is a mathematical formula that is used to find the distance between two points in a coordinate plane. It is derived from the Pythagorean theorem and is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where d is the distance between the two points, and (x1, y1) and (x2, y2) are the coordinates of the two points.

Q: How do we use the distance formula to find the abscissa of the other end of a line, given the length of the line and the coordinates of one end?

A: To find the abscissa of the other end of a line, we can use the distance formula and the given information. We know the length of the line (d = 13) and the coordinates of one end (x1 = 4, y1 = 8). We also know the ordinate (y2 = 3) of the other end. We can plug in the values into the distance formula and solve for the abscissa (x2) of the other end.

Q: What is the quadratic formula, and how is it used to solve quadratic equations?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: How do we use the quadratic formula to solve the quadratic equation in the problem?

A: To solve the quadratic equation in the problem, we can use the quadratic formula. We have the equation:

x2^2 - 8x2 - 128 = 0

We can plug in the values into the quadratic formula and solve for x2.

Q: What are the two possible values of x2, and how do we choose the correct value?

A: The two possible values of x2 are:

x2 = (8 + 24) / 2 = 32 / 2 = 16

x2 = (8 - 24) / 2 = -16 / 2 = -8

We choose the correct value of x2 based on the given information. Since the length of the line is 13 units, the x-coordinate of the other end must be greater than 4. Therefore, the correct value of x2 is 16.

Q: What are some real-world applications of the distance formula and the quadratic formula?

A: The distance formula and the quadratic formula have many real-world applications in various fields, such as:

  • Surveying: In surveying, we need to find the coordinates of points on the ground. The distance formula can be used to find the distance between two points.
  • Navigation: In navigation, we need to find the shortest distance between two points. The distance formula can be used to find the shortest distance.
  • Engineering: In engineering, we need to find the coordinates of points in a coordinate plane. The distance formula can be used to find the distance between two points.

Q: What are some tips for solving problems involving the distance formula and the quadratic formula?

A: Here are some tips for solving problems involving the distance formula and the quadratic formula:

  • Read the problem carefully: Make sure you understand what is being asked and what information is given.
  • Use the correct formula: Use the distance formula or the quadratic formula as needed to solve the problem.
  • Check your work: Make sure you have solved the problem correctly by checking your work.
  • Use real-world examples: Use real-world examples to help you understand the problem and to make it more interesting.

Conclusion

In this article, we have discussed the distance formula and the quadratic formula and how they are used to solve problems involving the distance between two points and quadratic equations. We have also discussed some real-world applications of these formulas and some tips for solving problems involving them.