### Midpoints in Triangles: Problem Set**Diagram Explanation:**The diagram shows a triangle $ \triangle ABC $ with points $ D $ and $ E $ marked on sides $ AB $ and $ AC $ respectively. Points $ D $ and $ E $ are midpoints of these sides, dividing $ AB $ and $ AC $ into two equal segments. The line segment $ DE $ runs parallel to $ BC $.**Problems:**1. **Given:** - $ BE = 2x $ - $ EC = x + 3 $ Find $ BC $.2. **Given:** - $ DE = 3x + 5 $ - $ AC = 4x - 10 $ Find $ AC $.3. **Given:** - $ AD = -6x + 14 $ - $ AB = 18x + 10 $ Find $ DB $.4. **Given:** - $ BC = 7x $ - $ EC = 4x - 9 $ Find $ BE $. ---**Tips for Solving:**- Use the property that $ DE $ is parallel to $ BC $ and $ D $ and $ E $ are midpoints to set up proportional relationships.- Remember that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long.---**Example Solution (a):**1. $ BE = 2x $2. $ EC = x + 3 $ Since $ E $ is the midpoint of $ AC $, $ AE $ is equal to $ EC $. Therefore, $ EC $ represents half of $ AC $. Substitute $ EC $ into the equation for $ EC $: \[ x + 3 = \frac{1}{2}AC \] To find $ BC $, consider the proportional relationship due to $ D $ and $ E $ being midpoints, giving $ BC $: \[ BC = 2BE \] Hence, \[ BC = 2(2x) = 4x \]Include these examples and tips as references for solving similar problems

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Answered: Dand E are the midpoints of the sides…

Math

Geometry

Dand E are the midpoints of the sides of A ABC. Find the following:

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

### Midpoints in Triangles: Problem Set

**Diagram Explanation:**
The diagram shows a triangle $ \triangle ABC $ with points $ D $ and $ E $ marked on sides $ AB $ and $ AC $ respectively. Points $ D $ and $ E $ are midpoints of these sides, dividing $ AB $ and $ AC $ into two equal segments. The line segment $ DE $ runs parallel to $ BC $.

**Problems:**

1. **Given:**
- $ BE = 2x $
- $ EC = x + 3 $

Find $ BC $.

2. **Given:**
- $ DE = 3x + 5 $
- $ AC = 4x - 10 $

Find $ AC $.

3. **Given:**
- $ AD = -6x + 14 $
- $ AB = 18x + 10 $

Find $ DB $.

4. **Given:**
- $ BC = 7x $
- $ EC = 4x - 9 $

Find $ BE $.

---

**Tips for Solving:**

- Use the property that $ DE $ is parallel to $ BC $ and $ D $ and $ E $ are midpoints to set up proportional relationships.
- Remember that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long.

---

**Example Solution (a):**

1. $ BE = 2x $
2. $ EC = x + 3 $

Since $ E $ is the midpoint of $ AC $, $ AE $ is equal to $ EC $. Therefore, $ EC $ represents half of $ AC $.

Substitute $ EC $ into the equation for $ EC $:
\[
x + 3 = \frac{1}{2}AC
\]

To find $ BC $, consider the proportional relationship due to $ D $ and $ E $ being midpoints, giving $ BC $:
\[
BC = 2BE
\]

Hence,
\[
BC = 2(2x) = 4x
\]

Include these examples and tips as references for solving similar problems

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