Are the triangles similar? If similar, state the theorem that proves they are similar. If not similar, write NONE. P 9 S 6 6 4 QAR
Are the triangles similar? If similar, state the theorem that proves they are similar. If not similar, write NONE. P 9 S 6 6 4 QAR
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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In the diagram provided, there are two triangles, \( \triangle PQS \) and \( \triangle QRS \):
1. Triangle \( \triangle PQS \):
- Side PS = 9
- Side PQ = 4
- Side QS = 6
2. Triangle \( \triangle QRS \):
- Side QR = 4
- Side RS = 6
- Side QS = 6
To determine if these triangles are similar, we can use the Side-Side-Side (SSS) Similarity Theorem. The SSS Similarity Theorem states that if the corresponding sides of two triangles are proportional, then the triangles are similar.
Let’s examine the ratios of the corresponding sides:
\[
\frac{PQ}{QR} = \frac{4}{4} = 1
\]
\[
\frac{QS}{RS} = \frac{6}{6} = 1
\]
\[
\frac{PS}{PR} = \frac{9}{6} = 1.5
\]
Since the corresponding sides are not proportional (\(\frac{PS}{PR} \neq 1\)), the triangles \( \triangle PQS \) and \( \triangle QRS \) are not similar.
Therefore, the answer is: **NONE**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F73f4c673-3324-4821-9039-a5ae2b13cf4f%2F2e6a37f6-de4e-43e3-9e71-6558f080a5ac%2F93z4frl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Are the triangles similar? If similar, state the theorem that proves they are similar. If not similar, write NONE.
In the diagram provided, there are two triangles, \( \triangle PQS \) and \( \triangle QRS \):
1. Triangle \( \triangle PQS \):
- Side PS = 9
- Side PQ = 4
- Side QS = 6
2. Triangle \( \triangle QRS \):
- Side QR = 4
- Side RS = 6
- Side QS = 6
To determine if these triangles are similar, we can use the Side-Side-Side (SSS) Similarity Theorem. The SSS Similarity Theorem states that if the corresponding sides of two triangles are proportional, then the triangles are similar.
Let’s examine the ratios of the corresponding sides:
\[
\frac{PQ}{QR} = \frac{4}{4} = 1
\]
\[
\frac{QS}{RS} = \frac{6}{6} = 1
\]
\[
\frac{PS}{PR} = \frac{9}{6} = 1.5
\]
Since the corresponding sides are not proportional (\(\frac{PS}{PR} \neq 1\)), the triangles \( \triangle PQS \) and \( \triangle QRS \) are not similar.
Therefore, the answer is: **NONE**.
![### Finding the Perimeter of Triangle QRS
In this exercise, we are tasked with determining the perimeter of triangle \( \triangle QRS \). The steps involve using the given information about the triangle, specifically the side lengths and angles, to calculate its perimeter. Please round your final answer to the nearest tenth if necessary. Please note that the figures presented are not necessarily drawn to scale.
#### Diagram Explanation
##### Left Triangle ( \( \triangle NOP \) ):
- Ideally used as a reference triangle.
- Side length \( NO \): 51 units
- Side length \( OP \): 57 units
- Side length \( PN \): 71 units
- Angle \( N \): 53 degrees
- Angle \( O \): 82 degrees
- Angle \( P \): 45 degrees
##### Right Triangle ( \( \triangle QRS \) ):
- This is the primary triangle we are working with.
- Side length \( QR \): 23 units
- Side length \( RS \): 25.7 units
- Side length \( SQ \): \( x \) (unknown)
- Angle \( Q \): 53 degrees
- Angle \( R \): 82 degrees
- Angle \( S \): 45 degrees
### Steps to Find the Perimeter of \( \triangle QRS \):
1. **Identify Given Measurements**:
- \( QR = 23 \) units
- \( RS = 25.7 \) units
2. **Calculate Unknown Side \( SQ (x) \)**:
As it is necessary to find the length of side \( SQ \) to determine the perimeter, we can use the Law of Cosines due to the given angles and side lengths.
3. **Use the Law of Cosines** (if applicable) to find \( SQ \):
Unfortunately, without additional information or context allowing us to determine side \( SQ \) directly or by further trigonometric relationships, we should consider learning additional steps or information here. (A practical step could involve splitting the triangle further or using other known relationships.)
4. **Find Perimeter**:
- Once \( SQ \) is known, we add up all side lengths:
\[
\text{Perimeter} = QR + RS + SQ
\]
5. **Round the Answer** to the nearest tenth if necessary.
This completes the organized approach to finding the perimeter of \( \triangle Q](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F73f4c673-3324-4821-9039-a5ae2b13cf4f%2F2e6a37f6-de4e-43e3-9e71-6558f080a5ac%2F6c4bnuj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Perimeter of Triangle QRS
In this exercise, we are tasked with determining the perimeter of triangle \( \triangle QRS \). The steps involve using the given information about the triangle, specifically the side lengths and angles, to calculate its perimeter. Please round your final answer to the nearest tenth if necessary. Please note that the figures presented are not necessarily drawn to scale.
#### Diagram Explanation
##### Left Triangle ( \( \triangle NOP \) ):
- Ideally used as a reference triangle.
- Side length \( NO \): 51 units
- Side length \( OP \): 57 units
- Side length \( PN \): 71 units
- Angle \( N \): 53 degrees
- Angle \( O \): 82 degrees
- Angle \( P \): 45 degrees
##### Right Triangle ( \( \triangle QRS \) ):
- This is the primary triangle we are working with.
- Side length \( QR \): 23 units
- Side length \( RS \): 25.7 units
- Side length \( SQ \): \( x \) (unknown)
- Angle \( Q \): 53 degrees
- Angle \( R \): 82 degrees
- Angle \( S \): 45 degrees
### Steps to Find the Perimeter of \( \triangle QRS \):
1. **Identify Given Measurements**:
- \( QR = 23 \) units
- \( RS = 25.7 \) units
2. **Calculate Unknown Side \( SQ (x) \)**:
As it is necessary to find the length of side \( SQ \) to determine the perimeter, we can use the Law of Cosines due to the given angles and side lengths.
3. **Use the Law of Cosines** (if applicable) to find \( SQ \):
Unfortunately, without additional information or context allowing us to determine side \( SQ \) directly or by further trigonometric relationships, we should consider learning additional steps or information here. (A practical step could involve splitting the triangle further or using other known relationships.)
4. **Find Perimeter**:
- Once \( SQ \) is known, we add up all side lengths:
\[
\text{Perimeter} = QR + RS + SQ
\]
5. **Round the Answer** to the nearest tenth if necessary.
This completes the organized approach to finding the perimeter of \( \triangle Q
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