Question
Line \( A B \) and point \( C \) are divided into two lines with a length ratio of \( 7: 5 \). If the difference in length of the segments is 3 cm , find the length of segment \( A B \).
Ask by Nguyen Mullins. in Mongolia
Jan 23,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The length of segment \( AB \) is 18 centimeters.
Solution
To solve the problem, let's interpret the given information step by step.
1. **Understanding the Division:**
- The line segment \( AB \) is divided by point \( C \) into two parts, \( AC \) and \( CB \).
- The ratio of the lengths \( AC : CB = 7 : 5 \).
2. **Expressing the Lengths:**
- Let the length of \( AC = 7k \).
- Then, the length of \( CB = 5k \).
3. **Difference in Lengths:**
- The difference between \( AC \) and \( CB \) is given as 3 cm.
\[
AC - CB = 7k - 5k = 2k = 3 \text{ cm}
\]
4. **Solving for \( k \):**
\[
2k = 3 \Rightarrow k = \frac{3}{2} = 1.5 \text{ cm}
\]
5. **Finding the Length of \( AB \):**
- \( AB = AC + CB = 7k + 5k = 12k \)
\[
AB = 12 \times 1.5 \text{ cm} = 18 \text{ cm}
\]
**Final Answer:** The length of segment \( AB \) is **18 centimeters**.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Bonus Knowledge
To find the length of segment \( AB \), let's denote the lengths of segments \( AB \) and \( C \) as \( 7x \) and \( 5x \) respectively, according to the given ratio of \( 7:5 \). The difference between these two segments is given as \( 3 \) cm. We can set up the equation based on the difference in lengths: \[ 7x - 5x = 3 \] This simplifies to: \[ 2x = 3 \] Now, solving for \( x \) gives: \[ x = \frac{3}{2} \text{ cm} \] Next, we can find the length of segment \( AB \): \[ AB = 7x = 7 \times \frac{3}{2} = \frac{21}{2} = 10.5 \text{ cm} \] Therefore, the length of segment \( AB \) is \( 10.5 \) cm.
