<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

Identify Accurate Drawings of Triangles

Whether a triangle is correct based on law of cosines.

0%
Progress
Practice Identify Accurate Drawings of Triangles
Progress
0%
Identify Accurate Drawings of Triangles

Your friend is creating a new board game that involves several different triangle shaped pieces. However, the game requires accurate measurements of several different pieces that all have to fit together. She brings some of the pieces to you and asks if you can verify that her measurements of the pieces' side lengths and angles are correct.

You take out the first piece. According to your friend, the piece has sides of length 4 in, 5 in and 7 in, and the angle between the side of the length 4 and the side of length 5 is . She's very confident in the lengths of the sides, but not quite sure if she measured the angle correctly. Is there a way to determine if your friend's game piece has the correct measurements, or did she make a mistake?

It is indeed possible to determine if your friend's measurements are correct or not. At the end of this Concept, you'll be able to tell your friend if her measurements were accurate.

Watch This

James Sousa: The Law of Cosines: Applications

Guidance

Our extension of the analysis of triangles draws us naturally to oblique triangles. The Law of Cosines can be used to verify that drawings of oblique triangles are accurate. In a right triangle, we might use the Pythagorean Theorem to verify that all three sides are the correct length, or we might use trigonometric ratios to verify an angle measurement. However, when dealing with an obtuse or acute triangle, we must rely on the Law of Cosines.

Example A

In at the right, , And . Is the drawing accurate if it labels as ? If not, what should measure?

Solution: We will use the Law of Cosines to check whether or not is .

Since , we know that is not . Using the Law of Cosines, we can figure out the correct measurement of .

For some situations, it will be necessary to utilize not only the Law of Cosines, but also the Pythagorean Theorem and trigonometric ratios to verify that a triangle or quadrilateral has been drawn accurately.

Example B

A builder received plans for the construction of a second-story addition on a house. The diagram shows how the architect wants the roof framed, while the length of the house is 20 ft. The builder decides to add a perpendicular support beam from the peak of the roof to the base. He estimates that new beam should be 8.3 feet high, but he wants to double-check before he begins construction. Is the builder’s estimate of 8.3 feet for the new beam correct? If not, how far off is he?

Solution: If we knew either or , we could use trigonometric ratios to find the height of the support beam. However, neither of these angle measures are given to us. Since we know all three sides of , we can use the Law of Cosines to find one of these angles. We will find .

Now that we know , we can use it to find the length of .

Yes, the builder’s estimate of 8.3 feet for the support beam is accurate.

Example C

In , and . Find the measure of all three angles.

Solution:

Guided Practice

1. Find using the Pythagorean Theorem, Law of Cosines, trig functions, or any combination of the three.

2. Find using the Pythagorean Theorem, Law of Cosines, trig functions, or any combination of the three if , and .

3. Use the Law of Cosines to determine whether or not the following triangle is drawn accurately. If not, determine how far the measurement of side "d" is from the correct value.

Solutions:

1. First, find .

2. (these two angles are a linear pair). . This simplifies to the quadratic equation . Using the quadratic formula, we can determine that . So, since .

3. To determine this, use the Law of Cosines and solve for to determine if the picture is accurate. , which means in the picture is off by 1.9.

Concept Problem Solution

Since your friend is certain of the lengths of the sides of the triangle, you should use those as the known quantities in the Law of Cosines and solve for the angle:

So as it turns out, your friend is rather close. Her measurements were probably slight inaccurate because of her round off from the protractor.

Explore More

1. If you know the lengths of all three sides of a triangle and the measure of one angle, how can you determine if the triangle is drawn accurately?

Determine whether or not each triangle is labelled correctly.

Determine whether or not each described triangle is possible. Assume angles have been rounded to the nearest degree.

1. In , b=4, c=4, d=5, and .
2. In , a=7, b=4, c=9, and .
3. In , b=3, c=2, d=7, and .
4. In , a=8, b=6, c=13.97, and .
5. In , a=4, b=4, c=9, and .
6. In , b=3, c=5, d=4, and .
7. In , a=8, b=3, c=6, and .
8. If you use the Law of Cosines to solve for in where a=3, b=7, and c=12, you will an error. Explain why.

To view the Explore More answers, open this PDF file and look for section 5.3.

Vocabulary Language: English

law of cosines

law of cosines

The law of cosines is a rule relating the sides of a triangle to the cosine of one of its angles. The law of cosines states that $c^2=a^2+b^2-2ab\cos C$, where $C$ is the angle across from side $c$.
The text passage provided is a lesson on using the Law of Cosines to verify the accuracy of triangle drawings. It includes examples and guided practice questions. However, for the purpose of the task, I will focus on the content structure and analysis of the passage itself.

### Content Structure:
The passage is a lesson on using the Law of Cosines to verify the accuracy of triangle drawings. It includes an introduction to the problem, examples, and guided practice questions. The examples and practice questions involve determining if given measurements of triangles are accurate or not.

### Analysis:
no_question

### explanation:
1. **content structure**:
   - the passage introduces the problem of verifying the accuracy of triangle drawings using the law of cosines.
   - it provides examples where the law of cosines is used to check the measurements of triangles.
   - the examples include step-by-step solutions to determine if the given measurements are correct.
   - the passage also includes guided practice questions for further practice.

2. **analysis**:
   - the passage does not contain a single mathematical question or a set of questions that are explicitly stated as problems to be solved by the reader.
   - instead, it provides examples and practice questions that are part of a larger lesson on the law of cos
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

Identify Accurate Drawings of Triangles

Whether a triangle is correct based on law of cosines.

0%
Progress
Practice Identify Accurate Drawings of Triangles
Progress
0%
Identify Accurate Drawings of Triangles

Your friend is creating a new board game that involves several different triangle shaped pieces. However, the game requires accurate measurements of several different pieces that all have to fit together. She brings some of the pieces to you and asks if you can verify that her measurements of the pieces' side lengths and angles are correct.

You take out the first piece. According to your friend, the piece has sides of length 4 in, 5 in and 7 in, and the angle between the side of the length 4 and the side of length 5 is . She's very confident in the lengths of the sides, but not quite sure if she measured the angle correctly. Is there a way to determine if your friend's game piece has the correct measurements, or did she make a mistake?

It is indeed possible to determine if your friend's measurements are correct or not. At the end of this Concept, you'll be able to tell your friend if her measurements were accurate.

Watch This

James Sousa: The Law of Cosines: Applications

Guidance

Our extension of the analysis of triangles draws us naturally to oblique triangles. The Law of Cosines can be used to verify that drawings of oblique triangles are accurate. In a right triangle, we might use the Pythagorean Theorem to verify that all three sides are the correct length, or we might use trigonometric ratios to verify an angle measurement. However, when dealing with an obtuse or acute triangle, we must rely on the Law of Cosines.

Example A

In at the right, , And . Is the drawing accurate if it labels as ? If not, what should measure?

Solution: We will use the Law of Cosines to check whether or not is .

Since , we know that is not . Using the Law of Cosines, we can figure out the correct measurement of .

For some situations, it will be necessary to utilize not only the Law of Cosines, but also the Pythagorean Theorem and trigonometric ratios to verify that a triangle or quadrilateral has been drawn accurately.

Example B

A builder received plans for the construction of a second-story addition on a house. The diagram shows how the architect wants the roof framed, while the length of the house is 20 ft. The builder decides to add a perpendicular support beam from the peak of the roof to the base. He estimates that new beam should be 8.3 feet high, but he wants to double-check before he begins construction. Is the builder’s estimate of 8.3 feet for the new beam correct? If not, how far off is he?

Solution: If we knew either or , we could use trigonometric ratios to find the height of the support beam. However, neither of these angle measures are given to us. Since we know all three sides of , we can use the Law of Cosines to find one of these angles. We will find .

Now that we know , we can use it to find the length of .

Yes, the builder’s estimate of 8.3 feet for the support beam is accurate.

Example C

In , and . Find the measure of all three angles.

Solution:

Guided Practice

1. Find using the Pythagorean Theorem, Law of Cosines, trig functions, or any combination of the three.

2. Find using the Pythagorean Theorem, Law of Cosines, trig functions, or any combination of the three if , and .

3. Use the Law of Cosines to determine whether or not the following triangle is drawn accurately. If not, determine how far the measurement of side "d" is from the correct value.

Solutions:

1. First, find .

2. (these two angles are a linear pair). . This simplifies to the quadratic equation . Using the quadratic formula, we can determine that . So, since .

3. To determine this, use the Law of Cosines and solve for to determine if the picture is accurate. , which means in the picture is off by 1.9.

Concept Problem Solution

Since your friend is certain of the lengths of the sides of the triangle, you should use those as the known quantities in the Law of Cosines and solve for the angle:

So as it turns out, your friend is rather close. Her measurements were probably slight inaccurate because of her round off from the protractor.

Explore More

1. If you know the lengths of all three sides of a triangle and the measure of one angle, how can you determine if the triangle is drawn accurately?

Determine whether or not each triangle is labelled correctly.

Determine whether or not each described triangle is possible. Assume angles have been rounded to the nearest degree.

1. In , b=4, c=4, d=5, and .
2. In , a=7, b=4, c=9, and .
3. In , b=3, c=2, d=7, and .
4. In , a=8, b=6, c=13.97, and .
5. In , a=4, b=4, c=9, and .
6. In , b=3, c=5, d=4, and .
7. In , a=8, b=3, c=6, and .
8. If you use the Law of Cosines to solve for in where a=3, b=7, and c=12, you will an error. Explain why.

To view the Explore More answers, open this PDF file and look for section 5.3.

Vocabulary Language: English

law of cosines

law of cosines

The law of cosines is a rule relating the sides of a triangle to the cosine of one of its angles. The law of cosines states that $c^2=a^2+b^2-2ab\cos C$, where $C$ is the angle across from side $c$.
user
The text passage provided is a lesson on using the Law of Cosines to verify the accuracy of triangle drawings. It includes examples and guided practice questions. However, for the purpose of the task, I will focus on the content structure and analysis of the passage itself.

### Content Structure:
The passage is a lesson on using the Law of Cosines to verify the accuracy of triangle drawings. It includes an introduction to the problem, examples, and guided practice questions. The examples and practice questions involve determining if given measurements of triangles are accurate or not.

### Analysis:
TYPE: NO_QUESTION

### Explanation:
1. **Content Structure**:
   - The passage introduces the problem of verifying the accuracy of triangle drawings using the Law of Cosines.
   - It provides examples where the Law of Cosines is used to check the measurements of triangles.
   - The examples include step-by-step solutions to determine if the given measurements are correct.
   - The passage also includes guided practice questions for further practice.

2. **Analysis**:
   - The passage does not contain a single mathematical question or a set of questions that are explicitly stated as problems to be solved by the reader.
   - Instead, it provides examples and practice questions that are part of a larger lesson on the Law of Cos