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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Topic Video
Question
![**Finding the Length of DF**
*Encuentra la longitud de DF*
In this problem, you are given two triangles, △ABC and △DEF. You are provided with the measurements of angles and sides, and are asked to find the length of side DF.
### Triangle △ABC:
- **Angle A:** 85°
- **Angle B:** 56°
- **Angle C:** 39° (since the sum of angles in any triangle is 180°)
- **Side AB:** 15 units
- **Side AC:** 20 units
- **Side BC:** 24 units
### Triangle △DEF:
- **Angle D:** 85°
- **Angle E:** 39°
- **Angle F:** 56°
- **Side EF:** 6 units
### Given Information:
- You are working with two triangles that have one corresponding angle pair each that are equal: △ABC and △DEF, implying that these triangles are **similar** due to Angle-Angle (AA) similarity criterion.
Since the triangles are similar, the corresponding sides of these triangles are proportional. That means:
\[
\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}
\]
### Applying the Proportion:
- **Sides for comparison:**
- \( DF \) compared to \( AC \)
- \( EF \) compared to \( BC \)
\[
\frac{DF}{AC} = \frac{EF}{BC}
\]
### Substituting the Known Values:
\[
\frac{DF}{20} = \frac{6}{24}
\]
### Solving for DF:
\[
\frac{DF}{20} = \frac{1}{4}
\]
\[
DF \times 4 = 20
\]
\[
DF = \frac{20}{4}
\]
\[
DF = 5 \text{ units}
\]
Thus, you are required to enter this value in the respective input box below the figures of the triangles.
\[
DF = 5 \text{ units}
\]
**Enter the length below:**
*Ingrese la longitud a continuación:*
\[
DF = \_\_\_\_ \text{ units}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6572212b-f9dd-4510-a4cf-9dc644f245bd%2F649beb1d-e002-48da-b2fc-237dd49d2738%2Fe2mrlpj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Length of DF**
*Encuentra la longitud de DF*
In this problem, you are given two triangles, △ABC and △DEF. You are provided with the measurements of angles and sides, and are asked to find the length of side DF.
### Triangle △ABC:
- **Angle A:** 85°
- **Angle B:** 56°
- **Angle C:** 39° (since the sum of angles in any triangle is 180°)
- **Side AB:** 15 units
- **Side AC:** 20 units
- **Side BC:** 24 units
### Triangle △DEF:
- **Angle D:** 85°
- **Angle E:** 39°
- **Angle F:** 56°
- **Side EF:** 6 units
### Given Information:
- You are working with two triangles that have one corresponding angle pair each that are equal: △ABC and △DEF, implying that these triangles are **similar** due to Angle-Angle (AA) similarity criterion.
Since the triangles are similar, the corresponding sides of these triangles are proportional. That means:
\[
\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}
\]
### Applying the Proportion:
- **Sides for comparison:**
- \( DF \) compared to \( AC \)
- \( EF \) compared to \( BC \)
\[
\frac{DF}{AC} = \frac{EF}{BC}
\]
### Substituting the Known Values:
\[
\frac{DF}{20} = \frac{6}{24}
\]
### Solving for DF:
\[
\frac{DF}{20} = \frac{1}{4}
\]
\[
DF \times 4 = 20
\]
\[
DF = \frac{20}{4}
\]
\[
DF = 5 \text{ units}
\]
Thus, you are required to enter this value in the respective input box below the figures of the triangles.
\[
DF = 5 \text{ units}
\]
**Enter the length below:**
*Ingrese la longitud a continuación:*
\[
DF = \_\_\_\_ \text{ units}
\]
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