text: <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 $c$.
explanation: The text extract contains a math question and examples that pertain to using the Law of Cosines to verify the accuracy of triangle measurements. It also includes a concept problem and solutions to guided practice questions.
  
  1. **Description of the Document:**
     - The document discusses the application of the Law of Cosines to verify the accuracy of measurements in oblique triangles.
     - It provides examples and solutions to problems where the Law of Cosines is used to check if given side lengths and angles of a triangle are correct.
     - The document includes a concept problem and guided practice questions with solutions.
  
  2. **Math Question and Answer:**
     - The concept problem asks whether a triangle with sides 4 in, 5 in, and 7 in, and an angle of 75 degrees between the sides of length 4 in and 5 in, is correctly measured.
     - The solution uses the Law of Cosines to determine that the given angle is not correct and calculates the correct angle.
  
  3. **Guided Practice Questions and Answers:**
     - The document includes several guided practice questions and their solutions, which involve using the Law of Cosines to verify triangle measurements and solve for unknown angles or sides.
  
  **
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: <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 $c$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract contains a math question and examples that pertain to using the Law of Cosines to verify the accuracy of triangle measurements. It also includes a concept problem and solutions to guided practice questions.
  
  1. **Description of the Document:**
     - The document discusses the application of the Law of Cosines to verify the accuracy of measurements in oblique triangles.
     - It provides examples and solutions to problems where the Law of Cosines is used to check if given side lengths and angles of a triangle are correct.
     - The document includes a concept problem and guided practice questions with solutions.
  
  2. **Math Question and Answer:**
     - The concept problem asks whether a triangle with sides 4 in, 5 in, and 7 in, and an angle of 75 degrees between the sides of length 4 in and 5 in, is correctly measured.
     - The solution uses the Law of Cosines to determine that the given angle is not correct and calculates the correct angle.
  
  3. **Guided Practice Questions and Answers:**
     - The document includes several guided practice questions and their solutions, which involve using the Law of Cosines to verify triangle measurements and solve for unknown angles or sides.
  
  **Label:**
  - The document contains a math