Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Question
![**Educational Content: Simplifying Triangles**
**Problem: Finding the Length of FG**
In the given problem, we need to find the length of the line segment FG. The diagram represents a triangle with the following details:
- \( \triangle ECG \) has a point D where the triangle is divided into two smaller triangles, \( \triangle EDF \) and \( \triangle DFG \).
- The lengths of the sides are marked as follows:
- \( ED = 24 \)
- \( DG = 54 \)
- \( EF = x \)
- \( FG = x+10 \)
**To Solve:**
1. **Identify Known Lengths:**
- \( ED = 24 \)
- \( DG = 54 \)
2. **Identify Variable Lengths:**
- \( EF = x \)
- \( FG = x + 10 \)
3. **Set up the Equation:**
- Recognize that \( D \) divides \( FG \) such that \( FG = ED + DG \).
Therefore:
\[ FG = 24 + 54 \]
\[ FG = 78 \]
4. **Solve for \( x \) using \( x + 10 = 78\):**
- Subtract 10 from both sides:
\[ x = 78 - 10 \]
\[ x = 68 \]
5. **Substitution:**
- Substitute \( x \) into \( FG = x + 10 \):
\[ FG = 68 + 10 \]
\[ FG = 78 \]
**Verification with Options Provided:**
- The possible answers are:
- 8
- 18
- 26
- 30
Since none of these options matches our calculated length of 78, it appears there might be an error in our analysis and needs reevaluation.
Alternatively, adjusting the approach or re-evaluating the equation setup might correct mismatches:
**Multiple Choice Options Check:**
Rechecking we realize our hypotheses should be based on geometry principles, such as similar triangles or Pythagorean It's good to check your exact angles or proportionality as potentially its application as is applying solving errors.
**Final Check**:
Without direct angle specifications ensure alignment with correct trigonometry and parameters:
As always ensure steps reintegrate using checking basic principles and reintegration.
**Educational Tip:**
Triangle problems can be](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb294f3d7-c37e-4d19-872e-5bb2e2baac27%2Fb29e2fb5-cee7-44a1-9bbb-0d6858c24a5f%2Fgu0astc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content: Simplifying Triangles**
**Problem: Finding the Length of FG**
In the given problem, we need to find the length of the line segment FG. The diagram represents a triangle with the following details:
- \( \triangle ECG \) has a point D where the triangle is divided into two smaller triangles, \( \triangle EDF \) and \( \triangle DFG \).
- The lengths of the sides are marked as follows:
- \( ED = 24 \)
- \( DG = 54 \)
- \( EF = x \)
- \( FG = x+10 \)
**To Solve:**
1. **Identify Known Lengths:**
- \( ED = 24 \)
- \( DG = 54 \)
2. **Identify Variable Lengths:**
- \( EF = x \)
- \( FG = x + 10 \)
3. **Set up the Equation:**
- Recognize that \( D \) divides \( FG \) such that \( FG = ED + DG \).
Therefore:
\[ FG = 24 + 54 \]
\[ FG = 78 \]
4. **Solve for \( x \) using \( x + 10 = 78\):**
- Subtract 10 from both sides:
\[ x = 78 - 10 \]
\[ x = 68 \]
5. **Substitution:**
- Substitute \( x \) into \( FG = x + 10 \):
\[ FG = 68 + 10 \]
\[ FG = 78 \]
**Verification with Options Provided:**
- The possible answers are:
- 8
- 18
- 26
- 30
Since none of these options matches our calculated length of 78, it appears there might be an error in our analysis and needs reevaluation.
Alternatively, adjusting the approach or re-evaluating the equation setup might correct mismatches:
**Multiple Choice Options Check:**
Rechecking we realize our hypotheses should be based on geometry principles, such as similar triangles or Pythagorean It's good to check your exact angles or proportionality as potentially its application as is applying solving errors.
**Final Check**:
Without direct angle specifications ensure alignment with correct trigonometry and parameters:
As always ensure steps reintegrate using checking basic principles and reintegration.
**Educational Tip:**
Triangle problems can be
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