The measures of the three sides of a triangle are AB = 30 in., AC = 50 in., and BC = 60in.. From a point D on side AB, a line DE is drawn through a point E on side AC such that angle AED is equal to angle ABC. If the perimeter of the triangle ADE is equal to 56 in., find the sum of the lengths of line segments BD and CE.

To solve this problem, we need to use similar triangles and the properties of a triangle's perimeter. Let's analyze the given triangle \( \triangle ABC \) and the smaller triangle \( \triangle ADE \) that is formed by the line segment \( DE \).

### Step-by-Step Solution:

1. **Given information**:
   - \( AB = 30 \) in.
   - \( AC = 50 \) in.
   - \( BC = 60 \) in.
   - \( P_{ADE} = AD + DE + AE = 56 \) in.
   - \( \angle AED = \angle ABC \) implies that \( \triangle ADE \sim \triangle ABC \) (angle-angle similarity).

2. **Determine the similarity ratio**:
   Since \( \triangle ADE \sim \triangle ABC \), the sides of \( \triangle ADE \) are proportional to the sides of \( \triangle ABC \).

   Let's denote the similarity ratio by \( k \). Thus, the sides of \( \triangle ADE \) can be expressed as:
   - \( AD = k \cdot AB \)
   - \( AE = k \cdot AC \)
   - \( DE = k \cdot BC \)

3. **Express the perimeter of \( \triangle ADE \) in terms of \( k \)**:
   The perimeter of \( \triangle ADE \) is given by:
   \[
   AD + DE + AE = k \cdot AB + k \cdot BC + k \cdot AC
   \]

   Substituting the known lengths:
   \[
   k \cdot 30 + k \cdot 60 + k \cdot 50 = 56
   \]

   Simplify this expression:
   \[
   k (30 + 60 + 50) = 56
   \]
   \[
   k \cdot 140 = 56
   \]

   Solve for \( k \):
   \[
   k = \frac{56}{140} = \frac{2}{5}
   \]

4. **Find \( AD \), \( DE \), and \( AE \)**:
   Using \( k = \frac{2}{5} \):
   - \( AD = k \cdot AB = \frac{2}{5} \times 30 = 12 \) in.
   - \( AE = k \cdot AC = \frac{2}{5} \times 50 = 20 \) in.
   - \( DE = k \cdot BC = \frac{2}{5} \times 60 = 24 \) in.

5. **Calculate \( BD \) and \( CE \)**:
   To find \( BD \) and \( CE \), we calculate the remaining parts of sides \( AB \) and \( AC \), respectively.
   - \( BD = AB - AD = 30 - 12 = 18 \) in.
   - \( CE = AC - AE = 50 - 20 = 30 \) in.

6. **Sum of \( BD \) and \( CE \)**:
   \[
   BD + CE = 18 + 30 = 48 \text{ in.}
   \]

### Conclusion:

The sum of the lengths of line segments \( BD \) and \( CE \) is **48 inches**.

---

Would you like to ask for details or have any questions?

Here are five related questions for further exploration:

1. How do you determine the similarity ratio between two similar triangles?
2. What is the significance of angle similarity in triangles?
3. Can you find the area ratio of \( \triangle ADE \) to \( \triangle ABC \)?
4. What would be the length of \( BE \) in this problem?
5. How can you use the similarity ratio to find the lengths of all segments of a triangle?

**Tip:** Always check if triangles are similar using the angle-angle (AA) criterion, as it can simplify finding unknown lengths.

Explore how to solve a triangle problem involving similar triangles and perimeter calculation. Understand the properties of triangles and how similarity affects side lengths. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation