7. Here is a rectangle with some right triangles inside it. a. Which triangles are similar to triangle AED? Explain or show your reasoning. and $$ riangle F E D $$ is similar to $$ A E D $$ because they are c=ght tsicongles. b. The length of segment EF is 8 units and the length of segment ED is 10 units. Find the length of segment FA. Explain or show your reasoning.

Question

7. Here is a rectangle with some right triangles inside it. a. Which triangles are similar to triangle AED? Explain or show your reasoning. and $$ 	riangle F E D $$ is similar to $$ A E D $$ because they are c=ght tsicongles. b. The length of segment EF is 8 units and the length of segment ED is 10 units. Find the length of segment FA. Explain or show your reasoning.

UpStudy AI · Accepted Answer

Certainly! Let's tackle each part of Problem 7 step by step.

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### **Problem 7**

**Given:** A rectangle containing some right triangles, including triangles $$ \triangle AED $$ and $$ \triangle FED $$.

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#### **Part a. Which triangles are similar to triangle $$ \triangle AED $$? Explain or show your reasoning.**

**Answer:**

**Triangles Similar to $$ \triangle AED $$:**

- **$$ \triangle FED $$:** 
  - **Reasoning:** Both $$ \triangle AED $$ and $$ \triangle FED $$ are right-angled triangles (each has a right angle). Additionally, they share a common acute angle at point $$ E $$. According to the **AA (Angle-Angle) Similarity Criterion**, if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Therefore, $$ \triangle FED $$ is similar to $$ \triangle AED $$.

- **Any other right triangles in the rectangle sharing the same acute angle:** 
  - If there are additional right triangles within the rectangle that share the acute angle at $$ E $$ (or any other corresponding acute angle with $$ \triangle AED $$), they would also be similar to $$ \triangle AED $$ by the same AA criterion.

**Summary:** $$ \triangle FED $$ is similar to $$ \triangle AED $$ because both are right triangles that share a common acute angle, satisfying the AA similarity condition.

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#### **Part b. The length of segment $$ EF $$ is 8 units and the length of segment $$ ED $$ is 10 units. Find the length of segment $$ FA $$. Explain or show your reasoning.**

**Answer:**

To find the length of segment $$ FA $$, we'll use the properties of similar triangles established in Part a.

**Given:**
- $$ \triangle FED \sim \triangle AED $$ (from Part a)
- $$ EF = 8 $$ units
- $$ ED = 10 $$ units

**Objective:** Find $$ FA $$.

**Approach:**
1. **Identify Corresponding Sides:**
   - In $$ \triangle FED $$ and $$ \triangle AED $$, the corresponding sides are proportional.
   - Let's map the sides as follows:
     - $$ FE $$ in $$ \triangle FED $$ corresponds to $$ AE $$ in $$ \triangle AED $$.
     - $$ ED $$ in $$ \triangle FED $$ corresponds to $$ AD $$ in $$ \triangle AED $$.
     - $$ FD $$ in $$ \triangle FED $$ corresponds to $$ ED $$ in $$ \triangle AED $$.

2. **Set Up the Proportion:**
   - Using the similarity ratio between corresponding sides:
     $$
     \frac{FE}{AE} = \frac{ED}{AD}
     $$
   - However, we need to find $$ FA $$. Assuming $$ FA $$ corresponds to $$ AE $$ in the similar triangles, the proportion becomes:
     $$
     \frac{EF}{FA} = \frac{ED}{AD}
     $$
   - But without knowing $$ AD $$, we need to relate $$ FA $$ directly using the given sides.

3. **Use the Pythagorean Theorem:**
   - Considering $$ \triangle FAE $$, which is also a right triangle (since it's part of the rectangle and shares the right angle at $$ E $$):
     $$
     FA^2 + EF^2 = AE^2
     $$
   - We need to express $$ AE $$ in terms of $$ ED $$ and the similarity ratio.

4. **Determine the Scale Factor:**
   - From the similarity $$ \triangle FED \sim \triangle AED $$:
     $$
     \text{Scale Factor} = \frac{EF}{AE} = \frac{8}{FA}
     $$
   - Given $$ ED = 10 $$ units corresponds to $$ AD $$ in the larger triangle:
     $$
     \frac{ED}{AD} = \frac{10}{AD}
     $$
   - Since the triangles are similar, the ratios of corresponding sides are equal:
     $$
     \frac{EF}{FA} = \frac{ED}{AD} \implies \frac{8}{FA} = \frac{10}{AD}
     $$
     - However, without $$ AD $$, we need another relationship.

5. **Assume Proportionality Based on Similar Triangles:**
   - If $$ \triangle FED $$ is scaled by a factor $$ k $$ to get $$ \triangle AED $$, then:
     $$
     FA = k \times EF \quad \text{and} \quad AD = k \times ED
     $$
   - From $$ \frac{EF}{FA} = \frac{ED}{AD} $$:
     $$
     \frac{8}{FA} = \frac{10}{AD} \implies \frac{8}{FA} = \frac{10}{k \times 10} = \frac{1}{k}
     $$
     - Therefore:
       $$
       k = \frac{FA}{8}
       $$
   - Combining with $$ AD = k \times 10 $$:
     $$
     AD = \left( \frac{FA}{8} \right) \times 10 = \frac{10 \times FA}{8} = \frac{5 \times FA}{4}
     $$

6. **Use $$ \triangle AED $$ Dimensions:**
   - Assuming the rectangle's sides are consistent, we could relate $$ FA $$ back using the Pythagorean theorem or proportionality.
   - However, without additional information about other sides of the rectangle or triangles, we simplify by considering the similarity ratio.

7. **Final Step:**
   - Given the side $$ EF = 8 $$ units in $$ \triangle FED $$ corresponds to $$ FA $$ in $$ \triangle AED $$, and $$ ED = 10 $$ units corresponds directly, we can set up the proportion based on similar triangles:
     $$
     \frac{EF}{FA} = \frac{ED}{AD} \implies \frac{8}{FA} = \frac{10}{AD}
     $$
   - If we assume that $$ AD $$ is proportional and solve for $$ FA $$, we derive:
     $$
     FA = \frac{8 \times AD}{10}
     $$
   - Without $$ AD $$, another approach is to recognize that in similar triangles, the ratios of corresponding sides are equal. If we consider $$ \triangle FED $$ scaled up by a factor $$ \frac{5}{4} $$ (since $$ \frac{10}{8} = \frac{5}{4} $$), then:
     $$
     FA = \frac{5}{4} \times FE = \frac{5}{4} \times 8 = 10 \text{ units}
     $$

**Conclusion:** The length of segment $$ FA $$ is **10 units**.

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**Note:** The exact relationship and final calculation heavily depend on the specific configuration of the triangles within the rectangle. If there are additional details or specific side relationships in your diagram, please provide them for a more accurate solution.
**Simplify Answer:** - **Part a:** $$ \triangle FED $$ is similar to $$ \triangle AED $$ because both are right triangles sharing a common acute angle. - **Part b:** The length of segment $$ FA $$ is **10 units**. **Summary:** $$ \triangle FED $$ is similar to $$ \triangle AED $$ due to their right angles and shared acute angle. Using the similarity ratio, the length of segment $$ FA $$ is determined to be 10 units.

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