please help ### Problem 29: Midpoints in Triangle Geometry**Problem Statement:**In triangle $ GHJ $ below, points $ A $, $ B $, and $ C $ are midpoints of the sides.![Triangle Diagram](image_url_placeholder) The diagram shows triangle $ GHJ $ with an inner triangle $ ABC $ formed by connecting the midpoints of the sides. Here, $ A $ is the midpoint of $ GH $, $ B $ is the midpoint of $ HJ $, and $ C $ is the midpoint of $ GJ $.**Given:**- $ AB = 3x + 8 $- $ GJ = 2x + 24 $**Question:**What is the length of $ AB $?**Options:**- 8- 14- 24- 28### Detailed Explanation of the Diagram:The diagram is essential for understanding the placement of points and segments in this midpoints problem:- The largest triangle is $ GHJ $.- Points $ A $, $ B $, and $ C $ are aligned such that each divides the sides $ GH $, $ HJ $, and $ GJ $ into equal segments.- The smaller triangle $ ABC $ is formed inside $ GHJ $ by connecting these midpoints.### Steps to Solve:1. **Understanding Midpoints:** - As $ A $ is the midpoint of $ GH $, $ B $ is the midpoint of $ HJ $, and $ C $ is the midpoint of $ GJ $, each mid-segment (e.g., $ AB $) is half the length of its corresponding base segment (in this case, $ GJ $).2. **Setting Up the Equation:** - Given that $ AB $ is half of $ GJ $: \[ AB = \frac{1}{2} GJ \] - Substituting the given expressions: \[ 3x + 8 = \frac{1}{2} (2x + 24) \]3. **Solving the Equation:** - Simplifying the right-hand side: \[ 3x + 8 = x + 12 \] - Rearranging terms to isolate \( x \

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Description: please help ### Problem 29: Midpoints in Triangle Geometry**Problem Statement:**In triangle $ GHJ $ below, points $ A $, $ B $, and $ C $ are midpoints of the sides.![Triangle Diagram](image_url_placeholder) The diagram shows triangle \( GHJ

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Answered: 29. In triangle GHJ below, points A, B…

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29. In triangle GHJ below, points A, B and C are midpoints of the sides. А G When AB 3x + 8 and GJ = 2x + 24, what is AB?

Elementary Geometry For College Students, 7e

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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 29: Midpoints in Triangle Geometry

**Problem Statement:**

In triangle $ GHJ $ below, points $ A $, $ B $, and $ C $ are midpoints of the sides.

![Triangle Diagram](image_url_placeholder)

The diagram shows triangle $ GHJ $ with an inner triangle $ ABC $ formed by connecting the midpoints of the sides. Here, $ A $ is the midpoint of $ GH $, $ B $ is the midpoint of $ HJ $, and $ C $ is the midpoint of $ GJ $.

**Given:**
- $ AB = 3x + 8 $
- $ GJ = 2x + 24 $

**Question:**
What is the length of $ AB $?

**Options:**
- 8
- 14
- 24
- 28

### Detailed Explanation of the Diagram:
The diagram is essential for understanding the placement of points and segments in this midpoints problem:

- The largest triangle is $ GHJ $.
- Points $ A $, $ B $, and $ C $ are aligned such that each divides the sides $ GH $, $ HJ $, and $ GJ $ into equal segments.
- The smaller triangle $ ABC $ is formed inside $ GHJ $ by connecting these midpoints.

### Steps to Solve:
1. **Understanding Midpoints:**
- As $ A $ is the midpoint of $ GH $, $ B $ is the midpoint of $ HJ $, and $ C $ is the midpoint of $ GJ $, each mid-segment (e.g., $ AB $) is half the length of its corresponding base segment (in this case, $ GJ $).

2. **Setting Up the Equation:**
- Given that $ AB $ is half of $ GJ $:
\[
AB = \frac{1}{2} GJ
\]
- Substituting the given expressions:
\[
3x + 8 = \frac{1}{2} (2x + 24)
\]

3. **Solving the Equation:**
- Simplifying the right-hand side:
\[
3x + 8 = x + 12
\]
- Rearranging terms to isolate \( x \

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