In figure, D and E are points on AB and AC respectively, such that DE ll BC. If AD = 

1/3BD, AE = 4.5 cm, find AC

In this problem, we're given a triangle \( \Delta ABC \), with points \( D \) and \( E \) lying on sides \( AB \) and \( AC \) respectively, such that \( DE \parallel BC \). We are also given:

- \( AD = \frac{1}{3} BD \)
- \( AE = 4.5 \, \text{cm} \)
  
We are asked to find \( AC \).

### Step-by-Step Solution:

1. **Use the Basic Proportionality Theorem (Thales' Theorem)**:
   Since \( DE \parallel BC \), by the Basic Proportionality Theorem, the segments along sides \( AB \) and \( AC \) are proportional. This means:
   \[
   \frac{AD}{BD} = \frac{AE}{EC}
   \]
   We are given \( AD = \frac{1}{3} BD \), which implies:
   \[
   \frac{AD}{BD} = \frac{1}{3}
   \]
   Hence:
   \[
   \frac{AE}{EC} = \frac{1}{3}
   \]

2. **Calculate the Length of \( EC \)**:
   We are given \( AE = 4.5 \, \text{cm} \). Using the ratio \( \frac{AE}{EC} = \frac{1}{3} \), we can express \( EC \) as:
   \[
   AE = \frac{1}{3} EC
   \]
   Therefore:
   \[
   EC = 3 \times AE = 3 \times 4.5 = 13.5 \, \text{cm}
   \]

3. **Find the Total Length of \( AC \)**:
   The total length of \( AC \) is the sum of \( AE \) and \( EC \):
   \[
   AC = AE + EC = 4.5 + 13.5 = 18 \, \text{cm}
   \]

Thus, the length of \( AC \) is \( 18 \, \text{cm} \).

---

Would you like further clarification on any part of the solution?

Here are five related questions for deeper understanding:
1. How does the Basic Proportionality Theorem apply in triangles with parallel lines?
2. If \( AD = \frac{1}{4} BD \), how would the length of \( AC \) change?
3. What if \( AE \) was unknown, and only \( AC \) was given? How would you solve it?
4. Can the Basic Proportionality Theorem be used in non-triangular shapes?
5. How would you prove that \( DE \parallel BC \)?

**Tip**: When dealing with proportionality in geometry, always check for parallel lines, as they often imply a proportional relationship between segments.

In figure, D and E are points on AB and AC respectively, such that DE ll BC. If AD = 1/3 BD, AE = 4.5 cm, find AC.

In this geometry problem, learn how to use the Basic Proportionality Theorem to find the length of AC when given AD = 1/3 BD and AE = 4.5 cm. This step-by-step solution breaks down the use of proportions in triangles, making it ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation