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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![### Geometry Problem on Triangle Perimeters
**Problem Statement:**
A, B, and C are midpoints of sides DF, DE, and FE respectively. If BC = 11, AC = 13, and AB = 15, find the perimeter of triangle DEF (ΔDEF).
**Given Diagram Description:**
The diagram displays a triangle DEF with:
- Triangles within it connecting midpoints A, B, and C to form a smaller inner triangle.
- Points A, B, and C are indicated to be the midpoints of the sides DF, DE, and FE respectively.
- Line segments BC, AC, and AB are highlighted.
**Solution:**
The perimeter of triangle DEF can be determined by noting that A, B, and C are midpoints, bisecting the sides of DEF. Given:
- BC = 11
- AC = 13
- AB = 15
Since A, B, and C are midpoints of DF, DE, and EF respectively, each segment of DEF is twice the corresponding segment of the smaller triangle ABC. Therefore:
- DF = 2 × AB = 2 × 15 = 30
- DE = 2 × AC = 2 × 13 = 26
- EF = 2 × BC = 2 × 11 = 22
The perimeter of ΔDEF:
\[ \text{Perimeter} = DF + DE + EF \]
\[ \text{Perimeter} = 30 + 26 + 22 \]
\[ \text{Perimeter} = 78 \]
Thus, the perimeter of triangle DEF is **78** units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F191e03d4-8930-41d1-b99a-4aa0b21d9c31%2Fe37d18ec-6f59-4c56-82f2-e143d561bcdb%2Fonhr7r8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem on Triangle Perimeters
**Problem Statement:**
A, B, and C are midpoints of sides DF, DE, and FE respectively. If BC = 11, AC = 13, and AB = 15, find the perimeter of triangle DEF (ΔDEF).
**Given Diagram Description:**
The diagram displays a triangle DEF with:
- Triangles within it connecting midpoints A, B, and C to form a smaller inner triangle.
- Points A, B, and C are indicated to be the midpoints of the sides DF, DE, and FE respectively.
- Line segments BC, AC, and AB are highlighted.
**Solution:**
The perimeter of triangle DEF can be determined by noting that A, B, and C are midpoints, bisecting the sides of DEF. Given:
- BC = 11
- AC = 13
- AB = 15
Since A, B, and C are midpoints of DF, DE, and EF respectively, each segment of DEF is twice the corresponding segment of the smaller triangle ABC. Therefore:
- DF = 2 × AB = 2 × 15 = 30
- DE = 2 × AC = 2 × 13 = 26
- EF = 2 × BC = 2 × 11 = 22
The perimeter of ΔDEF:
\[ \text{Perimeter} = DF + DE + EF \]
\[ \text{Perimeter} = 30 + 26 + 22 \]
\[ \text{Perimeter} = 78 \]
Thus, the perimeter of triangle DEF is **78** units.
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