Using the Number Line to Find the Point

Oct 13, 2025 by miodeya.com 56 views

The Point Dividing a Line Segment into a 1:2 Ratio

Understanding the Problem

In mathematics, a line segment is a part of a line that is bounded by two distinct points. The line segment can be divided into different ratios, and one of the most common ratios is the 1:2 ratio. In this problem, we are given two points, J and K, which are located at -9 and -3 on the number line, respectively. We need to find the point that divides the line segment JK into a 1:2 ratio.

Using the Number Line to Find the Point

To find the point that divides the line segment JK into a 1:2 ratio, we need to use the number line. The number line is a visual representation of the real numbers, and it can be used to solve problems involving ratios and proportions.

Step 1: Find the Length of the Line Segment JK

The first step is to find the length of the line segment JK. The length of a line segment is the distance between its two endpoints. In this case, the length of JK is the distance between points J and K.

Length of JK = |J - K| = |-9 - (-3)| = |-9 + 3| = |-6|

The length of JK is 6 units.

Step 2: Divide the Line Segment into a 1:2 Ratio

Now that we know the length of JK, we can divide it into a 1:2 ratio. A 1:2 ratio means that the line segment is divided into two parts, one part is 1 unit long, and the other part is 2 units long.

Let x be the length of the shorter part (1 unit).
Then, 2x is the length of the longer part (2 units).
Since the total length of JK is 6 units, we can set up the equation:
x + 2x = 6
Combine like terms:
3x = 6
Divide both sides by 3:
x = 2

The length of the shorter part is 2 units, and the length of the longer part is 4 units.

Step 3: Find the Point that Divides the Line Segment

Now that we know the lengths of the two parts, we can find the point that divides the line segment JK. Since the shorter part is 2 units long, we need to move 2 units from point J towards point K.

Point J is at -9.
Move 2 units from point J towards point K:
-9 + 2 = -7

The point that divides the line segment JK into a 1:2 ratio is -7.

Conclusion

In this problem, we used the number line to find the point that divides the line segment JK into a 1:2 ratio. We first found the length of the line segment JK, then divided it into a 1:2 ratio, and finally found the point that divides the line segment. The point that divides the line segment JK into a 1:2 ratio is -7.

Key Concepts

Line segment: a part of a line that is bounded by two distinct points.
Number line: a visual representation of the real numbers.
Ratio: a comparison of two numbers.
Proportion: a statement that two ratios are equal.

Real-World Applications

Architecture: when designing buildings, architects need to divide lines into different ratios to create symmetrical and aesthetically pleasing designs.
Engineering: engineers use ratios and proportions to design and build structures, such as bridges and buildings.
Art: artists use ratios and proportions to create balanced and harmonious compositions.

Practice Problems

Find the point that divides the line segment AB into a 3:4 ratio, where A is at -2 and B is at 5.
Find the point that divides the line segment CD into a 2:3 ratio, where C is at 1 and D is at 7.

Solutions

The point that divides the line segment AB into a 3:4 ratio is 1.
The point that divides the line segment CD into a 2:3 ratio is 4.
The Point Dividing a Line Segment into a 1:2 Ratio: Q&A

Understanding the Problem

In our previous article, we used the number line to find the point that divides the line segment JK into a 1:2 ratio. We first found the length of the line segment JK, then divided it into a 1:2 ratio, and finally found the point that divides the line segment. In this article, we will answer some common questions related to the problem.

Q: What is a line segment?

A: A line segment is a part of a line that is bounded by two distinct points. It is a straight line that has a definite length and is bounded by two endpoints.

Q: What is a number line?

A: A number line is a visual representation of the real numbers. It is a line that is marked with numbers, and it can be used to solve problems involving ratios and proportions.

Q: What is a ratio?

A: A ratio is a comparison of two numbers. It is a way of expressing the relationship between two quantities.

Q: What is a proportion?

A: A proportion is a statement that two ratios are equal. It is a way of expressing the relationship between two quantities in a more general way.

Q: How do I find the point that divides a line segment into a 1:2 ratio?

A: To find the point that divides a line segment into a 1:2 ratio, you need to follow these steps:

Find the length of the line segment.
Divide the line segment into two parts, one part is 1 unit long, and the other part is 2 units long.
Find the point that divides the line segment.

Q: What if the line segment is not divided into a 1:2 ratio?

A: If the line segment is not divided into a 1:2 ratio, you need to find the point that divides the line segment into the given ratio. To do this, you need to follow the same steps as before, but with the given ratio.

Q: Can I use the number line to find the point that divides a line segment into a 3:4 ratio?

A: Yes, you can use the number line to find the point that divides a line segment into a 3:4 ratio. The steps are the same as before, but with the given ratio.

Q: What if I have a line segment with a negative length?

A: If you have a line segment with a negative length, you need to follow the same steps as before, but with the negative length.

Q: Can I use the number line to find the point that divides a line segment into a 1:1 ratio?

A: Yes, you can use the number line to find the point that divides a line segment into a 1:1 ratio. The point that divides the line segment into a 1:1 ratio is the midpoint of the line segment.

Q: What if I have a line segment with a fractional length?

A: If you have a line segment with a fractional length, you need to follow the same steps as before, but with the fractional length.

Conclusion

In this article, we answered some common questions related to the problem of finding the point that divides a line segment into a 1:2 ratio. We hope that this article has been helpful in understanding the problem and its solution.

Key Concepts

Line segment: a part of a line that is bounded by two distinct points.
Number line: a visual representation of the real numbers.
Ratio: a comparison of two numbers.
Proportion: a statement that two ratios are equal.

Real-World Applications

Architecture: when designing buildings, architects need to divide lines into different ratios to create symmetrical and aesthetically pleasing designs.
Engineering: engineers use ratios and proportions to design and build structures, such as bridges and buildings.
Art: artists use ratios and proportions to create balanced and harmonious compositions.

Practice Problems

Find the point that divides the line segment AB into a 3:4 ratio, where A is at -2 and B is at 5.
Find the point that divides the line segment CD into a 2:3 ratio, where C is at 1 and D is at 7.

Solutions

The point that divides the line segment AB into a 3:4 ratio is 1.
The point that divides the line segment CD into a 2:3 ratio is 4.

Additional Resources

For more information on line segments and ratios, please see our previous article on the topic.
For more practice problems and solutions, please see our practice problems section.