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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![**Problem:**
*I is the midpoint of \( \overline{HJ} \) and F is the midpoint of \( \overline{GK} \). What is the value of \( x \)?*
**Diagram Explanation:**
- A circle is drawn with points H, J, G, and K on its circumference.
- Points I and F are indicated as the midpoints of segments \( \overline{HJ} \) and \( \overline{GK} \) respectively.
- Line segments \( \overline{IJ} \), \( \overline{HF} \), and \( \overline{EF} \) are drawn within the circle.
- Segment \( \overline{IE} \) is labeled with the expression \( x - 4 \).
- Segment \( \overline{EF} \) is labeled with the value 3.
- The segments \( \overline{HI} \) and \( \overline{GF} \) are marked to indicate they are equal due to being midpoints.
The goal is to determine the value of \( x \) based on the diagram and given information.
**Solution:**
- Since I is the midpoint of \( \overline{HJ} \), segment \( \overline{HI} = \overline{IJ} \).
- Similarly, since F is the midpoint of \( \overline{GK} \), segment \( \overline{GF} = \overline{FK} \).
Next, utilize the given lengths to set up an equation.
**Steps:**
1. Look at the segments on the triangle involving midpoint lengths.
2. If a segment is given two expressions due to its position, equate these expressions.
3. Solve for \( x \).
Given the lengths:
- \( \overline{IE} = x - 4 \)
- \( \overline{EF} = 3 \)
- Solve for \( x \).
\[ x - 4 + 3 = x - 1 \]
Complete the calculation step-by-step to determine \( x \).
The provided diagram gives us a visual representation, setting up corresponding lengths for calculating values based on given midpoint rules. Ensure the triangles formed by these midpoints follow standard geometric properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718438f1-000a-489a-b339-ddbb88468aee%2Fbdf0f43d-24e5-4b73-9449-d0c865926bae%2Flk0hh9mm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
*I is the midpoint of \( \overline{HJ} \) and F is the midpoint of \( \overline{GK} \). What is the value of \( x \)?*
**Diagram Explanation:**
- A circle is drawn with points H, J, G, and K on its circumference.
- Points I and F are indicated as the midpoints of segments \( \overline{HJ} \) and \( \overline{GK} \) respectively.
- Line segments \( \overline{IJ} \), \( \overline{HF} \), and \( \overline{EF} \) are drawn within the circle.
- Segment \( \overline{IE} \) is labeled with the expression \( x - 4 \).
- Segment \( \overline{EF} \) is labeled with the value 3.
- The segments \( \overline{HI} \) and \( \overline{GF} \) are marked to indicate they are equal due to being midpoints.
The goal is to determine the value of \( x \) based on the diagram and given information.
**Solution:**
- Since I is the midpoint of \( \overline{HJ} \), segment \( \overline{HI} = \overline{IJ} \).
- Similarly, since F is the midpoint of \( \overline{GK} \), segment \( \overline{GF} = \overline{FK} \).
Next, utilize the given lengths to set up an equation.
**Steps:**
1. Look at the segments on the triangle involving midpoint lengths.
2. If a segment is given two expressions due to its position, equate these expressions.
3. Solve for \( x \).
Given the lengths:
- \( \overline{IE} = x - 4 \)
- \( \overline{EF} = 3 \)
- Solve for \( x \).
\[ x - 4 + 3 = x - 1 \]
Complete the calculation step-by-step to determine \( x \).
The provided diagram gives us a visual representation, setting up corresponding lengths for calculating values based on given midpoint rules. Ensure the triangles formed by these midpoints follow standard geometric properties.
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