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What is the Division of a Line Segment in a Given Ratio?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The division of a line segment in a given ratio means finding a point on that line segment that splits it into two smaller parts, where the lengths of these two parts have a specific relationship (ratio). It's like cutting a piece of rope into two pieces of a certain proportion.
Simple Example
Quick Example
Imagine you have a 10 cm long 'laddoo' bar. You want to cut it so that one piece is twice as long as the other. This means you are dividing the laddoo bar in the ratio 2:1. The point where you cut it will be 6.67 cm from one end and 3.33 cm from the other.
Worked Example
Step-by-Step
Let's say you have a line segment AB that is 15 cm long. You want to divide it in the ratio 2:3.
Step 1: Understand the ratio. The ratio 2:3 means the line segment is divided into 2 parts and 3 parts, making a total of 2 + 3 = 5 equal parts.
---Step 2: Find the length of one 'part'. Since the total length is 15 cm and there are 5 parts, each part is 15 cm / 5 = 3 cm long.
---Step 3: Calculate the length of the first section. The first part of the ratio is 2, so the first section will be 2 * 3 cm = 6 cm long.
---Step 4: Calculate the length of the second section. The second part of the ratio is 3, so the second section will be 3 * 3 cm = 9 cm long.
---Step 5: Check your answer. The two sections are 6 cm and 9 cm. Their sum is 6 + 9 = 15 cm, which is the original length. The ratio of their lengths is 6:9, which simplifies to 2:3.
Answer: The line segment is divided into two parts, 6 cm and 9 cm long.
Why It Matters
Understanding ratios and line division is super important in many fields! Engineers use it to design bridges and buildings, ensuring parts are balanced. Data scientists use it to split data for analysis, and even in AI/ML, it helps in scaling features. It's a foundational skill for future innovators!
Common Mistakes
MISTAKE: Adding the ratio numbers and dividing the total length by only one part of the ratio. E.g., for 2:3, dividing 15 by 2 or 3. | CORRECTION: Add the ratio numbers (2+3=5) to find the total number of 'parts' the line segment is conceptually divided into. Then divide the total length by this sum.
MISTAKE: Reversing the ratio. If asked to divide in 2:3, they divide in 3:2. | CORRECTION: Pay close attention to the order of the ratio. 2:3 means the first part is proportional to 2, and the second part is proportional to 3.
MISTAKE: Not checking if the sum of the divided parts equals the original total length. | CORRECTION: Always add the lengths of the divided segments. This sum must be equal to the original length of the line segment.
Practice Questions
Try It Yourself
QUESTION: A 20-meter road needs to be divided into two sections for painting in the ratio 1:4. How long will each section be? | ANSWER: 4 meters and 16 meters
QUESTION: A school ground is 45 meters long. If a running track divides it in the ratio 2:7, what are the lengths of the two parts? | ANSWER: 10 meters and 35 meters
QUESTION: A 36 cm long wooden stick is divided into three pieces. The first two pieces are in the ratio 1:2, and the second and third pieces are in the ratio 2:3. What are the lengths of all three pieces? (Hint: The middle piece length is common to both ratios.) | ANSWER: 6 cm, 12 cm, 18 cm
MCQ
Quick Quiz
A 24 cm ribbon is divided in the ratio 3:5. What is the length of the shorter piece?
8 cm
9 cm
15 cm
12 cm
The Correct Answer Is:
B
The total parts are 3+5=8. Each part is 24/8 = 3 cm. The shorter piece corresponds to the '3' in the ratio, so its length is 3 * 3 = 9 cm.
Real World Connection
In the Real World
When city planners design roads or railways, they often need to divide land segments in specific ratios to balance traffic flow or land use. For example, dividing a road into pedestrian and vehicle lanes, or allocating space for different types of shops in a market complex, uses this concept.
Key Vocabulary
Key Terms
RATIO: A comparison of two numbers or quantities | LINE SEGMENT: A part of a line that has two distinct endpoints | DIVISION: The act of splitting something into parts | PROPORTION: A relationship between parts of a whole
What's Next
What to Learn Next
Great job learning about dividing line segments! Next, you can explore 'Coordinate Geometry' where you'll learn to find this division point on a graph using coordinates, which is super useful for making maps and designs.
