A Segment Has Endpoints At \[$(3, -4)\$\] And \[$(3, -17)\$\]. How Many Units Long Is The Segment?

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Introduction

In the coordinate plane, a segment is a part of a line that connects two points. Calculating the length of a segment is essential in various mathematical applications, including geometry, trigonometry, and physics. In this article, we will explore how to calculate the length of a segment given its endpoints.

What is a Segment?

A segment is a part of a line that connects two points. It is a line segment, which is a set of points that includes the two endpoints and all the points between them. The length of a segment is the distance between its endpoints.

Calculating the Length of a Segment

To calculate the length of a segment, we can use the distance formula. The distance formula is derived from the Pythagorean theorem and is used to find the distance between two points in a coordinate plane.

The Distance Formula

The distance formula is given by:

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

where dd is the distance between the two points, (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the coordinates of the two points.

Applying the Distance Formula

Let's apply the distance formula to the given segment with endpoints at (3,−4)(3, -4) and (3,−17)(3, -17).

Step 1: Identify the Coordinates

The coordinates of the two endpoints are:

  • (x1,y1)=(3,−4)(x_1, y_1) = (3, -4)
  • (x2,y2)=(3,−17)(x_2, y_2) = (3, -17)

Step 2: Plug in the Values

Now, let's plug in the values into the distance formula:

d=(3−3)2+(−17−(−4))2d = \sqrt{(3 - 3)^2 + (-17 - (-4))^2}

Step 3: Simplify the Expression

Simplifying the expression, we get:

d=(0)2+(−13)2d = \sqrt{(0)^2 + (-13)^2}

d=0+169d = \sqrt{0 + 169}

d=169d = \sqrt{169}

d=13d = 13

Conclusion

In this article, we calculated the length of a segment given its endpoints using the distance formula. We applied the formula to a specific example and simplified the expression to find the length of the segment. The length of the segment is 13 units.

Frequently Asked Questions

Q: What is the distance formula?

A: The distance formula is a mathematical formula used to find the distance between two points in a coordinate plane.

Q: How do I apply the distance formula?

A: To apply the distance formula, you need to identify the coordinates of the two points, plug in the values into the formula, and simplify the expression.

Q: What is the length of the segment with endpoints at (3,−4)(3, -4) and (3,−17)(3, -17)?

A: The length of the segment is 13 units.

Further Reading

If you want to learn more about the distance formula and its applications, you can check out the following resources:

References

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Introduction

In our previous article, we explored how to calculate the length of a segment given its endpoints using the distance formula. In this article, we will answer some frequently asked questions related to the distance formula and its applications.

Q&A

Q: What is the distance formula?

A: The distance formula is a mathematical formula used to find the distance between two points in a coordinate plane. It is given by:

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

Q: How do I apply the distance formula?

A: To apply the distance formula, you need to identify the coordinates of the two points, plug in the values into the formula, and simplify the expression.

Q: What is the difference between the distance formula and the Pythagorean theorem?

A: The distance formula is a variation of the Pythagorean theorem, which is used to find the length of the hypotenuse of a right triangle. The distance formula is used to find the distance between two points in a coordinate plane.

Q: Can I use the distance formula to find the distance between two points in 3D space?

A: Yes, you can use the distance formula to find the distance between two points in 3D space. However, you need to use the 3D distance formula, which is given by:

d=(x2−x1)2+(y2−y1)2+(z2−z1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

Q: How do I find the midpoint of a segment?

A: To find the midpoint of a segment, you need to use the midpoint formula, which is given by:

(xm,ym)=(x1+x22,y1+y22)(x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

Q: Can I use the distance formula to find the length of a segment with negative coordinates?

A: Yes, you can use the distance formula to find the length of a segment with negative coordinates. The distance formula works with both positive and negative coordinates.

Q: How do I find the distance between two points on a coordinate plane with different scales?

A: To find the distance between two points on a coordinate plane with different scales, you need to use the distance formula and take into account the scale factor.

Examples

Example 1: Finding the distance between two points with the same x-coordinate

Let's find the distance between the points (3,−4)(3, -4) and (3,−17)(3, -17).

d=(3−3)2+(−17−(−4))2d = \sqrt{(3 - 3)^2 + (-17 - (-4))^2}

d=(0)2+(−13)2d = \sqrt{(0)^2 + (-13)^2}

d=0+169d = \sqrt{0 + 169}

d=169d = \sqrt{169}

d=13d = 13

Example 2: Finding the distance between two points with different x-coordinates

Let's find the distance between the points (2,3)(2, 3) and (5,7)(5, 7).

d=(5−2)2+(7−3)2d = \sqrt{(5 - 2)^2 + (7 - 3)^2}

d=(3)2+(4)2d = \sqrt{(3)^2 + (4)^2}

d=9+16d = \sqrt{9 + 16}

d=25d = \sqrt{25}

d=5d = 5

Conclusion

In this article, we answered some frequently asked questions related to the distance formula and its applications. We also provided examples of how to use the distance formula to find the distance between two points in a coordinate plane.

Further Reading

If you want to learn more about the distance formula and its applications, you can check out the following resources:

References